First of all, a lot of the migration period is in doubt nowadays and research has not uncovered a general model. (Sadly, most of the archeology is also done in German and mostly published on the provincial level, so it is not accessible to the English world).

However, from my perusal of the evidence it was a mix of them being invited to settle there in return for military service (a typical Roman model that had worked well for over four centuries), available land for colonisation due to the Roman withdrawal from Britain (though the withdrawal is doubted and might not have happened on such a large scale as previously believed) and a general weakness on part of local defences (which had been handled by the Roman state until the fifth century). Imagine a world where the national guard and army withdraw from a state and said state then has to defend itself against invading armies.

Those people also had a long tradition in serving in the Roman military, so they knew the areas involved. For example, archeologists have unearthed remains of a former Roman soldier's homestead in Northern Germany - said man returned to his home after serving in the Army. He carried extensive Roman goods with him like belt buckles, chairs, dinner ware etc.

TL:Dr: a mix of all the reasons you listed.

Good books on the subject include this one, which should help you with a bibliography. Like I said, most of the archeology or recent research into why they left their homes is inaccesible to non-german speakers, but a collection of semi-recent research (thouhg with a naval focus) has been published under the title Maritime Celts, Frisians and Saxons (London 1990)

The best museum to visit on the subject of the North German coastal tribes (specifically saxons) and their daily life as well as their emigration is IMO [this one] (http://www.burg-bederkesa.de/index.html), should you ever find yourself in this area.

/u/anchist has given some great facts. I'd like to add a bit:

According to the Ecclesiastical History of the English People, written by Bede, a monk and scribe at Lindesfarne Monastery in the late 700s, the Angles and Saxons were INVITED to Britain by the indigenous peoples to help defend them against the natives of present day Scotland. During Roman times, the Romans defended the Britons from these peoples (Hadrian's wall, circa 200, was the border).

According to Bede, after the Romans left the isle the Britons fell under brutal attack by the Scots/Celts when Hadrian's wall was unmanned. They petitioned the Angles and Saxons of present Germany to come fight for them. The irony is that the Angles and Saxons became just as oppressive to them as the Scots/Celts ever were.

edit: punctuation.

Textually, the two main sources British sources for the period are Gildas' De excidio et conquestu Britanniae (c.540?) and later on, Bede's Greater Chronicle (c.725).

The problems with Gildas are numerous - he has no chronology and the little he has is wrong, few names to whom we can ascribe any certainty, we don't have a firm date for his writing as the only dating of his work comes from the 950s, he is contradicted by continental sources on the events, but he does indicate it's the Saxons who come under invitation by Vortigern, whom they then turn on and take over the island (a point confirmed by the Gallic Chronicle of 452 but which puts it at a later date). Some scholars (Sims-Williams) think that we can't use him for almost anything, but I'll take the less hyper-critical stance and say that what he says in broad might be supportable. Bede inherits Gildas information (along with a short reference from Constantius' Life of Germanus) but tries to fit him into the framework that he is familiar with, but retains the same nub- Saxons were invited but then took over. Both are writing religious and politically charged pieces, so their historical worth is problematic but not impossible. But the Romans did use mercenaries on the continental side - it could be that the British did the same.

As for the archaeological evidence for the Angles, Jutes, and Saxons, there's not very much to distinguish one from the other at least on the British side, so the problem is whether Bede's assertion that the Angles, Saxons, and Jutes all came over might be reasonably queried. There are no specific burial sites in Britain that can be linked back to specific groups on the continent - the materials are so varied, the only conclusion is that they're not British, not that they're from a specific place in Saxony or on the Elbe-Wesser triangle. They all have similarities with places on the continent, but continental material culture in the 4th century was already mixing/mixed, so it's hard to determine what exactly constitutes a Saxon in the archaeological record. Material culture in Britain is difficult to place - some places have heavier concentrations of a certain type of item than others, so a particular kind of brooch gets associated with a particular area, but for example, Kent shows evidence of 'Saxon' and 'Anglian' brooches as well as Scandinavian elements (Jutish) as well as Gallo-Roman and Frankish material.

In short, the literary sources say things happened that so far we can't attest to in the archaeological record with exactness. Anglo-Saxons certainly came over and changed the material culture, but exactly what group of people that was composed of can't be determined beyond 'somewhere on the continent' (Y chromosome variation between the Basque and the western parts of Britain, Germanic/Scandinavian for the east and southern parts) That's the conclusion of the most recent work with isotopic investigations, but that hasn't really been applied to a large enough skeleton sample to be very useful, and it's difficult to differentiate genetically between someone who arrived in 1AD and someone who arrived in 450AD. The DNA field is too unstructured to help at this point.

Terrific answers so far. To recap, the story of the Anglo-Saxon invasion in 450 was invented much much later, probably due to Bede's (730s) misreading of a sermon by Gildas (500s). Yes migration happened, but there was probably a continuous flow of migrants moving in all directions across the North Sea throughout the early middle ages. Serious scholars have largely rejected anything we thought we knew about Britain during the years 400-700 and are beginning to rewrite the story from the ground up, that is, through archaeology.

The story goes something like this. The Roman administration withdrew its interests from Britain sometime during the 400s. There were probably some militia units that remained, particularly in the area of Hadrian's Wall, but they were left to themselves. Payment and supply shipments ended, sending Britain into a sudden economic decline.

This was awful news for the elites, who no longer had the infrastructure to create massive villas and manage large populations of labor. It was mediocre news for the vast majority of the population, whose tax burden was dramatically lightened but who no longer had access to some important imperial products, like mass-produced nails and pottery.

As the empire disintegrated on the continent, people were moving everywhere, but people had been mobile under the empire as well. Archaeologists now think that some of the "militia" units defending Britain had actually been recruited from across the Rhine. Once the Roman supply system crashed, they began wearing things like they had back home. Some of them returned to the continent, others invited their families over.

Since the Romano-British aristocrats no longer had the infrastructures to support their expensive lifestyles, military leaders were now the closest thing that remained to an elite. Everything turned wild west, and some of the native British signed on with the militias, swapping their native gear for Saxon style. Things were complex, to say the least.

Without a Roman administration to hold Britain together, British society fragmented, and communities during the 500s and 600s were highly localized. But as new elites gradually coalesced in these centuries, they did so in part by banding into regional aristocratic networks. In a world with few roads, these networks were naturally clustered along the major river networks, especially the Thames, the Severn, and the Trent.

Fast-forward to Bede writing in his cell in the 730s. As he surveys what's become of Britain, he sees three distinct cultures, and they've all got these remembered stories about German ancestors. He divides them neatly into Angles, Saxons, and Jutes, and a story is born!

• This basic narrative above and my reasons for not trusting the texts follow (more or less) Guy Halsall's Worlds of Arthur (2013). Strongly recommended!

• For an even more grassroots approach that pretty much discards Bede altogether, see Robin Fleming's Britain after Rome (2010). Halsall explains the kings; Fleming examines the people.

• And if you're interested in peaking into the black box of archaeology and seeing how we can reconstruct some what was happening in these early communities (specifically my source for thinking about rivers as regions), see Tom Williamson's Sutton Hoo and its Landscape (2008).

If you're interested, I talk a bit more about Halsall and Fleming here and especially here. Cheers!