Many enslaved women in the late medieval Mediterranean were forced into work as wet nurses. This is an excerpt from a longer article I wrote on wet nursing in Latin Europe:

For enslaved women forced to serve as wet nurses, the situation was even more demeaning than that of the municipal nurses. Since 1179, canon law had forbidden Christians from hiring Jews and Muslims to nurse their children (although the frequency of reiterations—from Christian and Jewish authorities alike—points to the frequently-informal shared breastfeeding duties among Christian and Jewish women neighbors). And yet from the thirteenth century into the early modern era, slaves in the Christian Mediterranean were almost always purchased or captured Muslims. To Christian parents, the solution was simple: forced baptism.

And even beyond this indignity, evidence concerning baptized Muslim wet nurses indicates the growing tendency to classify and judge people by skin color. The humour and heat imbalances in darker-skinned women made them bad mothers, according to medical authorities. And since wet nurses were understood as little more than biological appendages of real mothers, over and over, slave purchase and rental contracts demonstrate a keen preference for light-skinned baptizatae of nursing age.

Yes, rental contracts. Enslaved women pressed into service as wet nurses weren’t always just serving the family they at least knew and were comfortable with. Owners might hire out a baptizata as a nurse for some extra cash, or sell her altogether if the price was attractive enough.

And if no wet nurse was available? The brutal, systematic, and endemic sexual exploitation of enslaved women in Iberia could be—was—turned to economic use. When the foundling hospital of Perpignan was so financially overwhelmed in 1456 that it appealed to the city for extra money, the rectors made it bleedingly clear that aristocratic men bringing in their own bastard children were to blame. One man in 1400 Barcelona openly admitted sending away his own child so the baby's mother, an enslaved woman, would be free to nurse his legitimate heir.

And therein lay the cruelest blow of medieval wet nursing practice. Medical thought and popular religious teaching alike forbade women from nursing more than one child at the same time. All these wet nurses, all these “mothers who weren’t,” were mothers. They were mothers who weaned their children too early and quickly in order to make some money for the family; they were mothers who lost a child in infancy and had milk but no one suckle; they were mothers sold away from their newborns forever.

(n.b. I'm super busy today so it will be a little while before I can get to follow-up questions if there are any; my apologies in advance for the delay.)

I'll let someone who is more knowledgeable about Mediterranean slavery answer some of this, but I can tackle a few of the other portions of your questions. But to be clear, my main area of knowledge is Britain. So while a lot of this can apply across Europe, keep in mind that I'm mostly talking about Britain.

That being said, Britain wasn't a cultural outlier, and during the height of the medieval slave trade it was very much tied in with the continent so it can provide some insight.

The Treatment of Muslim Slaves

Muslims (or Saracens as they called them) were treated as an "other" by the Anglo Saxons, but the relationship was complicated. On the one hand, you have Synods denouncing Saracen practices, but on the other hand, Anglo Saxon medicine included materials that could only be acquired from the Islamic world and you had pilgrims like Willibald travelling to the Caliphate and you even had corrupted Arabic script appearing on the coinage of King Offa of Mercia.

The relationship between the two groups was a weird one.

But that being said, for reasons that will become clear as we go on, it's unlikely that if there were Muslim slaves in Britain they would have been treated all that differently from other slaves. So I'll just focus on slavery in general.

Christian Slavery

In your question you said that you heard that "slavery mostly died out in medieval Europe among Christians."

That is half true, but it obscures a very important fact.

Slavery prospered for quite a long time under Christianity. It was practiced since the early days of the Conversion, and continued to be a very common practice through to about the 11th century. In fact, entire post-conversion cities based their economy on it.

Many times the Scandinavians take the blame for this practice, but that isn't factually accurate. Europeans were enslaving each other long before anyone came aviking. What the Scandinavians did was allow the European slave traders to tap into a much larger market, thanks to their extensive trade routes. And the West responded to these new markets with wild enthusiasm. In Britain, as an example (as that's what I'm most familiar with) we see records of enslavement going in every possible direction. Everyone was enslaving each other, or at least trying to. And that includes the Scandinavian "Vikings." The Irish records, in particular, speak of how they managed to defeat enslave Scandinavian settlements on the island.

So that's a long way of saying that this wasn't a "pagan" practice, and the spread of Christianity didn't lead to it dying out. In fact, even 500 years after the conversion we still see slave trading cities in Britain. Bristol, for example, continued to have a thriving slave market into the 11th century.

Becoming a Slave in Christian Society

Raiding and wars were a common way to be enslaved in Christian Europe, but not the only way.

Even medieval legal codes made it rather easy to find yourself enslaved. For example, in Wessex, if you were a thief your punishment would be a 60 shilling fine. No biggie. But if you brought the stolen goods home to your family... well, under the Laws of Ine you and your whole family would be enslaved. Permanently. This was a particularly brutal considering that a wife couldn't bar her husband from bringing things home. So if you were married to a thief, you were at tremendous risk. It also lead to strange outcomes like infants being enslaved for their father's crimes, which was something that really enraged Archbishop Wulfstan.

You could also find yourself enslaved because of a bad harvest. We see records of entire communities selling themselves into slavery to a wealthy landowner who just happened to have enough food to feed them (and given the practice of Food Rent, I can virtually guarantee that they grew the food they were given in exchange for their freedom... which is an extra-special level of awful). And again, these people were Christians.

Slavery was big business and everyone, including the Christians, wanted in on it.

The Rights of Slaves in Christian Society

Now, as for the rights of slaves. They pretty much had none. Eventually certain laws were enacted in England that regulated the practice... kind of.

For example, you could be ransomed out of slavery. However, the slave owner had the sole choice of whether or not to ransom you home and at what price. Furthermore, if your family couldn't or wouldn't pay your ransom then you were stuck. You couldn't ransom yourself because, unlike slaves under systems like the Code Noir, you couldn't work for wages yourself. You had no income, period.

As for your well being, that's rather ugly too. For the first year of your enslavement you have some protections. If you die during that first year, your family can demand a weregild (it means "man price" and is basically a wrongful death penalty). The thinking goes that for that first year your family had an interest in your life because they might be able to ransom you back, but after that year passes, they no longer can claim a weregild. Which means that the slave owner no longer has to worry about paying them an expensive penalty if he kills you. So good luck.

And if your slave master is mistreating you... Tough luck. You don't have legal standing to raise a complaint to the authorities. You're a slave.

Slaves could also be sold overseas, which was catastrophic because it meant you had pretty much no hope of ever being freed and getting home. Early on this seems to have been a rare occurrence, perhaps to deal with particularly unpopular members of the community. But with the rise of the scandinavian slave trading routes, suddenly there was a lot of money to be made by selling slaves overseas. So rather than it being a rare occurrence, it become common. It also made ransoming very difficult because now a small villager from an impoverished kingdom would have to compete on an open market with a vast trade network that included the frigging Byzantines.

The Social Status of Slaves in Christian Society

There was a heavy layer of judgment laid down upon slaves. In the early days, if you read of any complaints regarding slavery it was about how terrible it was that Christans had Pagan masters. They never complained about Christians owning Christians, or Christians owning Pagans. The crime they were complaining about wasn't the slavery, it was the paganism.

Well, as paganism came to an end in western Europe, they had a new line they were drawing. Worthy vs unworthy. Archbishops lamented how there were slaves who were "unforworhte". Basically, undeserving. Which, naturally, indicates that there were deserving slaves. And reading the letters and sermons you get a picture of who was undeserving of being enslaved. They were virtuous, but more importantly they were well born.

It follows, therefore, that the people who deserved slavery were pretty much everyone else who found themselves in that situation. So pretty much everyone who wasn't noble and completely innocent.

This means that in addition to having virtually no way to get out of being enslaved, and having no legal rights, and being completely at the mercy of your master... you also have everyone, including even the Archbishop, clucking their tongues and talking about how you deserve your fate.

As you can imagine, there wasn't any social mobility available for slaves outside of manumission, and they were treated pretty badly.

And again, these were Christians doing this.

The "medieval times" covers a pretty large area and a pretty large time frame. Slavery was still relatively common in the early medieval period, partially because it took a while for old roman practices to die out even after the empire fell apart, and partially because the use of slaves was already common in the germanic tribes that would come to politically dominate most of western Europe during the 5-10th centuries. Slavery in this period was not nearly on the same scale as roman slavery, as the slave market was tied to the large scale cash crop production oriented and trade economy of the roman world. After the collapse of rome, trade and the production of cash craps dropped dramatically. Most agriculture became more subsistence based and much smaller scale. The need for slaves was reduced as a requirement of the economic system, but they did not disappear. Additionally the roman period saw a huge stratification of the elite. Its estimated the top 1% of romans owned over 50% of all slaves. As the urban roman society declined in the 4-5th centuries, these insanely wealthy elites declined as well, and people spread out into the countryside, with less , again further reducing the need for slaves.

Because of this, the type of slavery changed. Household slaves became the most common type of slavery. Instead of massive slave plantations with hundreds of slaves serving an elite aristocrat, most wealthy households moved to personal house slaves for cooking, domestic chores, and other daily duties. Usually having between 1-4 slaves.

The Visigothic code from the 5-7th centuries states that if someone finds an abandoned child, raises it to adulthood, and then finds the original birth parents, the birthparents are required to either give the adopted parents a slave, or the equivalent price of slave to the adopted parents as payment for their service.

In Byzantium, there is lots of evidence for the continuation of slavery. The Book of the Eparch from the 900s list all of the jobs available for slaves, and it actually includes more jobs for slaves than was available for freemen. They are notably all semi professional craft type jobs. Usually when one of thinks of slaves, they think manual labor. But in Byzantium we see slaves as notaries, silk makers, merchant, perfume makers, candle makers, soap makers, grocers, saddlers, inn keepers, carpenters, locksmiths, painters...etc. In these positions, the slave would have had to been specially trained, likely sent by his master to a journeyman to learn the trade. It seems slaves might have been seen as more trustworthy, since slaves could be tortured or killed for misbehaving, while regular salaried workers could not be punished as severely. Many slaves in Byzantium were foreigners, as buying exotic humans from other places was a status symbol of the elite.

At the opposite side of Europe, we have Ireland, which was never directly a part of the Roman empire, so had its own legacy of slavery outside of the Roman one. Slavery here was not as exotic or expensive as in the Byzantine Empire. Many slaves here were poor peasants captured during raids by warriors or pirates. Many women were captured to use as domestic slaves, and there was a definite sexual exploitation of female slaves here that is mentioned in lots of literature. Male slaves usually filled an agricultural role in Ireland. Herders, shepherds, plowman, threshing, woodcutting...etc. Its likely that in Ireland, a wealthy agricultural household might own one or two male slaves that were used as general labor in a "farm hand" sense, and were used for any manual labor needed around the farm.

A famous example of Irish Slavery is St Patrick. He was captured in the late 5th century by Irish pirates. He was only 16, and spent the next 5 or so years as a slave tending the flocks of a local irishman. His job was to watch the herd at night and protect it from wolves and predators. One can imagine that this was a rough life. He likely would have spent large stretches alone outside with the flocks, provided only what was necessary for his basic survival as far as food, clothing, and shelter went. During his later years, he strongly condemns the practice of slavery, so it is quite clear he did not have a positive experience of it during his time as a slave. In his own accounts in "Confessio" he says he endured "hunger and nakedness daily".... I have doubt that he was actually naked for his entire six years in the cold and harsh weather of Ireland, but it certainly doesnt paint a pleasant picture of his treatment.

Even by the Norman period, slavery in northern Europe was still relatively common. The Domesday book written in 1086 by William the first in England lists almost 10% of the entire population of England as a slave.

References consulted: Slavery after Rome, 500-1100 by Alice Rio

A follow up question. What about the Ottoman empire? The word “slave” originates from this periods, since the turks were enslaving slavic people, so probably worth mentioning along with the others.

Follow up question: Were Christians more likely to have been enslaved in (North African / Asian) Muslim countries?

I've read that some North African City States / nations raided Western coastal communities as far North as Ireland. is this true?

This recent post on slavery vs. serfs from u/sowser is one of the best I've read at AskHistorians, and I think it covers a lot related to the question you are asking.

Can someone recommend one or several good texts on medieval slavery? This is something I know very little about in comparison to colonial slavery. I'd like to read more.