Most sources for the early dealings with the Haudenosaunee and other Iroquois tribes are in French. As a result, I have not studied these relations as closely as others, but I can certainly share what I do know and have sources for!

If you do happen to speak French, I highly recommend the journals and writings of Gabriel Sagard, and Louis Nicolas. Both were French missionaries who lived with various Iroquois tribes in the seventeenth century. Further, The Jesuit Relations and Allied Documents, published in the mid nineteenth century, consists of a compilations of Jesuit journals and notations written and compiled in the seventeenth century. It is available online here: http://puffin.creighton.edu/jesuit/relations/ and is utterly fascinating.

Mourning wars were not always necessarily to capture or "replace" a lost tribesmember. Sometimes the goal was simply revenge - to kill the offending enemy on a one-for-one basis. When a captured individual was intended to be assimilated, the process would vary depending on the similarities of culture. Sagard wrote in one of his entries about a man captured from a neighboring tribe for whom the assimilation process and beatings were essentially symbolic. The man ran through a gauntlet of the tribe who beat him with sticks, but Sagard noted that they did not seem to be intending to cause injury, which contrasts with several other examples of downright mutilation.

Some cases are far more gruesome, and the intent seems to be twofold. Firstly they are in many ways mentally breaking the individual to allow them to be re-created in the vision of the adopting tribe, and secondly to only take strong individuals who do not break under pressure. People could be beaten and tortured, with some methods especially cruel. One Jesuit writer described the practice of driving arrows longways down the length of a man's forearm. If they screamed, they were killed. If they bore the pain, they were seen as fit for adoption. Another priest noted starvation tactics employed, and that people kidnapped in mourning wars would often not be fed until they had sufficiently broken down mentally.

How well these people were treated in the long term varies wildly too. Some writers noted that some of these "slaves" seemed to be genuinely treated as members of the tribe. Another notes that when a young Eskimo man, aged 23 and captured several years before was sick, the tribe simply left him to his own devices, not making any attempts to cure him. He was described as "so skinny his bones poked through his skin." Other assimilated tribesmen were noted to be "doing their work with fingerless-hands," suggesting that while they may be brought into the tribe, the process of doing so was brutal.

Individuals taken would often be placed into the role of the individual they were supposed to replace. A young man replacing a fallen warrior, (if assimilated) would take his place and be considered the new son of the fallen man's mother, take any wives the fallen man had had, etc. A woman captured to replace a fallen woman would take her spot just as closely, becoming the new wife of whomever had lost theirs. I recall reading that there was significant social pressure to treat the new individual exactly like the lost one, but I cannot for the life of me find the passages.

Europeans were treated harsher in general, but with exception. It also may simply be that their culture was too different. Several were simply tortured to death. The Jesuit logs are rife with examples of men burnt to death, or slowly skinned alive, etc. One man was described upon his rescue as having had "all of the flesh of his forearms stripped from the bone." Then only several paragraphs later, a French woman captured and assimilated was noted to have been allowed religious freedom and allowed to openly pray.

As /u/TreeofMadrigal has already explained, the process for adopting individual war prisoners from a Mourning War often began with considerable violence. Prior to adoption a prisoner might have his or her hands maimed (fingernails or finger joints removed) or be cut on the shoulder in order to make it more difficult to escape the restraints he or she wore while being led back to Iroquoia. The warrior who captured the prisoners earned prestige for the accomplishment, but did not have the authority to determine the prisoner's fate, though it was no unheard of for someone to take matters into his own hands rather than wait for an official ruling. There were incidents in which those too weak to travel, either due to age, illness, or injury, were killed before being adopted.

The leader of the war party divided the prisoners up among the participating clans or nations. The war party could make recommendations regarding whether a prisoner was suitable for adoption, and this was often conveyed through the colors they painted the faces of the prisoners when they arrived at their destination. Ideal adoptees were painted white, those of uncertain quality received red painted, and those doomed to execution were painted black. Women and children usually received white paint; non-Iroquoian (Algonquian and European) men more often received black paint. When running the gantlet, captives painted black received the harshest treatment while those in white might run through while only have clubs waved in their general direction.

Ultimately though, the women waiting in the villages would make the official ruling. The women divided out their clan's share of the captives to grieving families. Prior to the mid-1600s, during a proper Mourning War, there were generally fewer captives than grieving families so only the most prominent families actually received one.

At this point the captive became the family's responsibility, to do with as they felt best. Among the old, ill, or male captives, some might still be killed if the family's grief still demanded blood. The doomed captive, like the adoptee, often went by the name of the deceased relative their replacing, and generally, provided the captive played his or her (usually his) part, they were well treated. The difference comes in later, when the captive is the guest of honor at a farewell feast. In normal circumstances, farewell feasts were held by the elderly who believed their deaths were imminent. The family afterward stripped the captive of his adoptive identity and began the torturous execution.

The main factor in determining if a family would adopt or execute a captive was the the proximity of their grief. If the death that prompted their participation in the Mourning War was recent or sudden, their captive would likely meet an untimely end; if the family's grief had some time to abate naturally, then they'd usually be more willing to adopt the captive. They were still the property of the adopting family for several years during the acculturating process, and for various reasons might still be executed. In a particularly dramatic, though not really representative, example, a Haudenosaunee chief known to the French as Nero (actual name and rank unknown) was reported to have had eighty captives under his care executed once he heard his brother had been killed in battle. In a case of the Haudenosaunee being on the receiving end of such an execution, the Haudenosaunee-Erie War began in earnest when the sister of a recently deceased Erie ambassador had Anenraes, an Onondaga leader captured by the Erie, executed after promises were given that he would be adopted.

The threat that an adopted captive might still be executed was the stick that encouraged acculturation, along with lesser punishments for disobedience. The carrot was the generally good treatment captives received for being well-behaved and useful members of Haudenosaunee society. The adoptee became the deceased's heir, inheriting their name and perhaps their social privileges if they were found to be deserving of such ranks. There are cases of Wendat (Huron) leaders become Haudenosaunee leaders after their capture and adoption, and in a particular famous case, the Kieuneka / Gaonageh of the Attiwendaronks (The Neutral Confederacy) was captured by the Haudenosaunee. She was the leader of the women's council and a descendant of Jigonsaseh, the woman who aided Haiwatha and the Great Peacemaker in founding the Haudenosaunee. Though the descendants of Jigonsaseh once lead the Haudenosaunee's own women's council, that line had recently died out among them. The Kieuneka was put on the fast-tract of adoption so that she could take over that vacant spot.

Most adoptees, though, were destined for more modest for more modest lives. Many were initially regulated to menial labor, such as serving as porters for hunters and traders. This is how they became equated with slaves by French observers, but they were less slaves and more the lowest ranking members of a family to whom such duties were usually given, captive or no. As they spent more time among the Haudenosaunee and became increasingly integrated into the culture, they might earn additional privileges suitable for their effort and talents. The adopting family was also responsible for teaching the adoptee the local language and instructing the adoptee in the local culture. After about four years, the adoptee was generally allowed to go about as he or she pleased and had no more reason to fear an expected execution or punishment for his or her behavior (at least, no more so than any other member of society). At this point marriages were used to further cement the adoptee's position in Haudenosaunee society.

During the Beaver Wars, of the middle through late 17th Century, the Haudenosaunee had a vast glut of captives (which is how Nero could have had eighty captives to execute to avenge his brother). When a small family was captured, they were split up and adopted by different Haudenosaunee clans. When a whole clan segment was captured, they were permitted to establish their own longhouse in among the people that had captured them. In some cases, adopted men were forbidden from joining war parties against their homeland for fear that they might attempt to escape; in other cases, they were encouraged to do so as a test of loyalty. Successful adoptees appear to have been dispatched back to their homelands as emissaries, encouraging their relatives to voluntarily join the Haudenosaunee. As the war against the Wendat continued, increasing numbers of them found heeded the advice of these adopted relatives. Soon large numbers of potential adoptees were flooding into Iroquoia.

A large number of Arendarhonon, one of the nations that made up the Wendat Confederacy, voluntarily joined the Onondaga (mostly members of an large anti-French, anti-Christian traditionalist faction). Another of the Wendat nations, the Tahontaenrat, joined the Seneca in sufficient numbers that they established their own town, Gandourgarae, in Seneca territory rather than being split up and incorporated into various Seneca clans. The Tahontaenrat were eventually mixed into the Seneca population through marriages. In 1668, the Jesuits reported that nearly 2/3 of the Oneida were naturalized adoptees, with similar numbers at Gandagaron, one of the major Mohawk towns. These proportionally large numbers of adopted captives, refugees, and immigrants allowed for some aspects of their original culture and language to survive for some time before blending in with the Haudenosaunee culture at large.

In the 18th Century, the flow of captives once again receded to a trickle, but the success of integrating large numbers of foreign nations into the Haudenosaunee prompted new policies. After defeating the Susquehannocks for control of the Susquehanna River, the Haudensaunee found they did not have the manpower to hold the territory themselves. They're population was too limited to populate the Susquehanna Valley and the Ohio Valley simultaneously. While the Ohio became the focus of the Haudenosaunee proper (especially the Seneca and Cayuga, many of whom had recent Wendat ancestors to the point that Europeans in the region found it hard to find someone who spoke 'proper' Seneca or Cayuga among "the Mingo"--as the cultural blend in the region was often called). The Susquehanna Valley, on the other hand, the Haudenosaunee attempted to populate with refugee communities drawn up from the Carolinas and Virginia. These included the Tutelo, Nanticoke, and--most famously--the Tuscaroras (who were given land on the upper-most portion of the Susquehanna, neighboring the Haudenosaunee nations themselves). These nations were offered various protections under Haudenosaunee law and expected to aid the Haudenosaunee in common defense of their shared territory, but they didn't receive representation at council (though they could lobby representatives from the Five Nations). After the American Revolution, the Tuscarora would earn full representation on the Haudensaunee's councils. These adopted nations were allowed to maintain their own languages and traditions, which is why the last speakers of Tutelo--a Siouan language from southwestern Virginia--lived at Grand River with the Six Nations. The Cayuga specifically responsible for the Tutelo adoption and promised that the Tutelo wouldn't die out while they were under the Haudenosaunee's protection, which is why there's a long-standing tradition of Cayuga being adopted into Tutelo families.