Not quite your question but close.

When Mehmet the II conquered Constantinople he moved a large number of his Ottoman populace to fill out the population as many of them moved away when the Ottomans took it over.

Also he encouraged those who were conquered to stay and they were treated well. This was due to their skills being sought as useful and more important than their race or culture. It was a really interesting time and event.

This is an interesting question and one that deserves a decent answer. I'm afraid I can't meet AskHistorians' high standards for a top level comment, but since no-one else has really responded, I'll give it a go, and if anyone wants to come along and tell me I've got it all wrong they're welcome.

There are some general factors limiting migration in the Middle Ages which I'll mention, after which I'll move on to talking about Western Europe and particularly England, which is the area I know the most about.

So the first points are that travel was generally lot more difficult, dangerous and expensive than in, say, the industrial or modern ages. Information was a lot more limited as well. Reports of conditions in another region are probably unreliable and out-of-date, and you probably won't be able to learn the local language, customs, laws, etc. until you get there. These things would also be a lot more localised, rather than the situation we have today where most people in a country can communicate in some shared language.

That said, authorities generally exercise a lot less control over their territories than they do today. So there was more scope for nomadic groups like gypsies, bedouin or steppe tribes. The concepts of "nations" and "countries" were not the same as they are now; individuals and groups would not necessarily "belong" to either - instead they might be part of a tribe, subjects of a lord, citizens of a city-state or have any number of often overlapping statuses that affected their ability to move around. Moving between a country might not be a problem, but if buying land isn't legal, how are you going to move to the next village?

Looking more specifically at Western Europe, a lot would depend on who you were, and the circumstances at the time. If you were a peasant you were tied to your land and your lord. If you left you could be forcibly returned. When you get there, are you going to be able to find land to farm, or somewhere to live? How are you going to pay for the trip? That said, periods of labour shortage, like after the black death, could change this situation. A village and a noble with land going uncultivated could welcome immigrants.

What if you were an artisan or a merchant or something like that? If you're a free man you're not tied to a lord. You'd be better able to afford a relocation, and your money would probably be welcomed. However you might come up against the guild system, designed to maintain standards and reduce competition. You might be able to buy a house in a new town, but if the weavers' guild or the merchants' guild stopped you practising your trade you wouldn't get very far.

What about a noble or a knight? You owe fealty to your lord or king, so legally you can't just leave them. Perhaps even more importantly, most of your wealth and status comes from your land. You might be able to find a sympathetic noble to take you on, but if they don't have some lands they're willing to grant you, you're not going to have the status or income you had before. Your most likely reason to move would probably be if you were granted new lands or a particular position by your liege, probably following a war.

All that said, there weren't the sort of formal, bureaucratic immigration controls we're familiar with. If you were a tinker, a travelling minstrel or a mercenary, then borders between countries probably wouldn't mean much to you. However you'd always be at risk of the local authorities deciding you were undesirable and kicking you out of the village or refusing you entry to the town.

So to summarise the answers to your questions:

• in England, at least, there was not an "immigration process" as we'd understand it today, but most people were legally tied to land and their lords, and might have a difficult legal status if they did move. You'd be very much at the mercy of local attitudes.
• I can't give you any figures or examples, but not many people moved far, except in particular circumstances, like wartime.