There is not a great deal of information on the day to day lives of the men who were missionaries in this time that we can firmly establish as fact. Most of what we know of these people come from inevitably biased sources such as hagiographies and the histories that some of these clergy men wrote. Of course these sources cannot be trusted to give us a full and accurate account of everything that went down, but they are largely what we are stuck with.

Most of what i can speak to relates to the conversion of the Anglo-Saxons and the later conversion of the Norse, but of course there were different practices in each, and there were other areas of heavy missionary activity at the same time.

Let's start with how the religion was introduced. In the case of the Anglo-Saxons, according to Bede, Christianity was brought back to Great Britain by a mission trip from the continent that landed in Kent. The current king, again according to Bede, had a Christian wife and he allowed the Christian mission to exist though he himself did not convert immediately. Eventually he did, and while he did not proscribe the old heathen ways, he favored the new religion more heavily and gave greater powers to the Church which then began to preach openly and draw large amounts of converts.

From there the faith spread to other kingdoms, and within a century or so of the founding of the mission in Kent the Anglo-Saxon kingdoms were Christianized. This often happened as a condition of peace between kingdoms or through the direct conversion of the king by a bishop or other missionary. In some other places, Saxony and Scandinavia, often miracles are attributed to be the cause for conversion. Adam of Bremen mentions this explicitly.

The missionaries often did "stick around" as you said, and often ended up as bishops or other important church officials in the areas that they had been ministering in. This is also born out in the conversion of Scandinavia as well as in England.

If you want essentially a "guide to conversion" looking through a translation of Bede's History of the English and the dialogue between Bishop Augustine and Pope Gregory would be an excellent place to start.

Sources:

Adam of Bremen, Gesta Hammaburgensis Ecclesiae Pontificum

Bede, Historia Ecclesiastica Gentis Anglorum, a good translation of this would also make for good further reading if you wanted to know what the Pope said about converting people and practices to be established afterwards.

for Scandinavia specifically there's a lot of interesting articles in Garipzanov's collection titled Conversion and Identity in the Viking Age