"I don't believe they could just move to the other kingdom and start cultivating the land"

Short Answer

At the very start of the medieval period, there was nothing stopping someone from trekking across Europe to set up a new farm if they wished it. The bureaucracy and legal institutions did not exist to even have a migration policy, let alone enforce one. Surviving law codes and archaeological evidence from Bavaria, the Netherlands, and Wales suggests that until the 8th century, most countries didn't bother regulating migration. Over time, as the bureaucratic capability of states improved and feudal models of society took root, it became harder to migrate legally because it required the permission of the nobility. This does mean that in order to migrate legally, all you had to do was get written permission from your lord.

Migration Period - 7th century

In a lot of places they could just set up a farm, though it depended on when and where you were and what you wanted to do. We have pretty solid archaeological evidence that long-distance migration was rather common, particularly among traders. The Byzantine Empire had no objection to people migrating in or out of their territory, and chemical analysis of skeletons has shown that even Wales was home to a small community of Greek settlers (https://www-sciencedirect-com.ezproxy01.rhul.ac.uk/science/article/pii/S030544031300023X). Similar analysis of burial sites in Bavaria show that there was migration to the region throughout the early medieval period. Chemical analysis of skeletons in Frisia (part of the Frankish kingdom) show that there were many immigrants from Scandinavia and southern England present in the region.

Laws from the very early Middle Ages (we're talking 6th century here) make no mention of migration, and since this was the tail end of the 'Migration Period', we can suppose that if someone wanted to move there was little to stop them both in a physical and legal sense - if something isn't illegal, it's legal. The Frankish law code (https://sourcebooks.fordham.edu/source/salic-law.asp) is fragmentary, but the Visigothic law code (http://www.documentacatholicaomnia.eu/03d/0506-0506,_AA_VV,Leges_Romanae_Visigotorum[Scott_JP_Curatore],_EN.pdf) exists in full and contains the following clause which suggests that it was not uncommon for people to move to other countries: "if the witnesses, or their relatives, or friends, should either be oppressed with age or infirmity, or resident in a foreign or distant province... then they shall assemble in that province." The significant migrant populations found in skeletal remains, combined with the lack of laws about migration, suggests that it really was as simple as moving to the other kingdom and renting/clearing some land for cultivation. The only issue might be if you settled on someone else's land.

But "Early Middle Ages" also includes the 8th to 10th centuries, when such legislation was passed and enforced, so let's talk about that for a bit.

700 - 1000 AD

One's freedom to move depended heavily on legal status. A growing number of people at that time were serfs, who were bound to their land and lord and were effectively slaves of their lord. They could try to move, but the law generally required that they be returned upon discovery, and if their lord wanted them back. Carolingian law states the following: "serfs dwelling on the domain of another, on being required by their former lord, shall not be given to him except for the former place; where it was first seen that they had lived, thither they shall be returned, and diligent inquiry shall be made concerning their status and the status of their relations." However, the comment about "diligent inquiry" suggests that some people assumed false identities and had families under them, which means that it could take years for a runaway serf to be discovered. But the phrase "being required by their former lord" implies that illegal migration could be made legal by the serf's original lord. Another Carolingian law suggests that some lords were quite relaxed about the movements and activities of their serfs. Clause 7 of the Capitulary at Diedenhofen, 821 AD, singles out the rulers of Flanders and Brabant as arch offenders in this regard and urged them to take more action against such 'conspiracy of the serfs' or be fined. I mention this because, even though there were differences between legal and illegal movement, a lot of lords turned a blind eye to "illegal" migration and it was not that hard to move "illegally". I put "illegal" in because if the lord didn't chase up runaways, it was seen as legal enough.

But let's look at your specific scenario of moving from Anglo-Saxon England to the continent, because that one is unusual. Both the Anglo-Saxons and the Carolingians had rather advanced legal codes for the time, and codes that are decently preserved. They also had rigid systems of serfdom and liege-lords which made migration difficult. A law of Edward the Elder (901-924) states: "Let no man receive another man's man without his leave whom he before followed, and until he be blameless towards every hand. If any one do so, let him make bot of my oferhyrnes (this means give compensation)." This means that, so long as someone had no outstanding debts and the lord gave them permission, they could leave for other lands. Another of his laws explicitly acknowledges a significant population of foreigners: "If any one wrong an ecclesiastic or a foreigner, through any means, as to money or as to life, then shall the king or the earl there in the land, and the bishop of the people, be unto him in the place of a kinsman and of a protector, unless he have another; and let bot be strictly made, according as the deed may be, to Christ and to the king, as it is fitting; or let him avenge the deeds very deeply who is king among the people." If you were completely free and financially independent, a merchant for example, then there was no law stopping you from selling your house in Trier and moving to London. If you were a serf or had given a feudal oath, then you needed to get the permission of your lord to move. This could be granted or refused at their discretion. It would then be possible to request to enter the service of another lord (as a lot of fighters did), either directly or through a bailiff, or find employment on a farm or in a city.