I believe that this 2006 paper by Mark G Thomas, Michael P.H Stumpf, and Heinrich Härke, and the new 2022 study by Gretzinger et al cover the controversies the best, although the 2006 paper itself was also controversial, and the newest study that you have linked to in your comment also appears to disagree with it.

To address your question directly (i.e. what exactly are people arguing about when it comes to migration into Britain in the early medieval period), the answer is essentially how it happened, and what happened after their arrival.

As the new study by Gretzinger et al says:

”The end of the Roman administration in fifth century Britain preceded a dramatic shift in material culture, architecture, manufacturing and agricultural practice, and was accompanied by language change”.

”From the Renaissance to the present day, the primary explanatory narrative for these changes has been invasion and conquest followed by resettlement from the continent. On the basis of a small set of written sources, it was supposed that the local Romano-British population was largely replaced by migrants from the Germanic-speaking part of the continent”.

And as the 2006 paper says:

”The traditional model of the Anglo-Saxon immigration into fifth-century Britain was based on scanty written sources (Gildas, Bede, Anglo-Saxon Chronicle) and envisaged mass ‘invasion’ from the Continent and large-scale replacement of the natives”.

As they both say, the older (perhaps “traditional”) theory of the Anglo-Saxon migrations into Britain posited that it was a large-scale invasion from the continent, that took place over a relatively brief period of time, and that it involved a colossal mass-replacement and slaughter of the native Britons.

This was largely based on two sources, a 6th century monk named Gildas, and a 7/8th century monk named Bede.

Gildas claimed in his writings that, following the Roman withdrawal, the Britons invited the Saxons into Britain in order to help defend it, but that the Saxons, frustrated with their lack of monthly supplies, eventually betrayed them and began to conquer the island.

This conquest, according to Gildas, involved a mass slaughter of the Britons in modern-day England, and a mass-fleeing of any survivors either overseas or into the mountains and the woods.

”For the fire of vengeance, justly kindled by former crimes, spread from sea to sea, fed by the hands of our foes in the east, and did not cease, until, destroying the neighbouring towns and lands, it reached the other side of the island”.

”Some [Britons] therefore, of the miserable remnant, being taken in the mountains, were murdered in great numbers; others, constrained by famine, came and yielded themselves to be slaves for ever to their foes, running the risk of being instantly slain, which truly was the greatest favour that could be offered them: some others passed beyond the seas."

”Others, committing the safeguard of their lives, which were in continual jeopardy, to the mountains, precipices, thickly wooded forests, and to the rocks of the seas (albeit with trembling hearts), remained still in their country”.

This description however is where the controversies start, as neither Gildas nor Bede are considered to be very historically accurate, and as the 2022 paper says:

”Such discrepancies between the archaeological record and historical narratives could be argued to support a rejection of migration or invasion hypotheses”.

For a time after this, the theory would change. Rather than a model of a massed invasion, a theory of a small-scale migration of a Germanic-speaking elite would become popular amongst historians and archaeologists such as Hodges in 1989 and Higham in 1992, eventually becoming “the majority opinion among Anglo-Saxonist scholars” according to the 2006 paper.

However, by 1998 this model would become controversial too, and in 2001 a genetic study calculated that between 24.4% and 72.5% of the English gene pool is made up of southern Danish and northern Germanic DNA.

If the actual number was to be as high as 72.5%, that would, according to the 2006 paper,:

”require migration on a massive scale (approx. 500 ,000+), well above documented population movements of the early Middle Ages”.

Hence why the issue has become so controversial.

Gildas’s description of a mass slaughter and exodus of the Britons doesn’t really correspond with the archeological or material culture, or the historical evidence, and a 2015 study found that Danish, north German, and northwestern German DNA makes up only around 10-40% of the modern English gene pool.

As the 2022 paper says though:

”populations change over time through drift and gene flow, so present-day populations may be poor proxies for ancient groups of unknown genetic makeup”.

The counter-hypothesis to a large-scale invasion over a small period of time was, like I said, a model of a small-scale, elite migration. However as the 2022 paper says:

”the available isotopic and DNA evidence, even if hitherto small scale, suggests that immigrants were less wealthy and buried alongside locals, which does not fit a model of elite influence”.

Thus, to quote the 2022 paper again:

”To this day, little agreement has been reached over the scale of migration, the mode of interaction between locals and newcomers, or how the transformation of the social, material, and linguistic or religious spheres was achieved”.

This 2022 paper found that, through a study of genome-wide ancient DNA from 460 medieval northwestern Europeans (including 278 individuals from England), that individuals from eastern England derived up to 76% of their ancestry from the continental North Sea.

Their answer for this though, rather than being one of the

”previous hypotheses about the social mechanisms in this migration [which] have included partial social segregation, elite migration, substantial population replacement or no migration at all”,

was a “regionally contingent migration with partial integration that was probably dependent on the fortunes of specific families and their individual members” that took place over a very long period of time (from during Roman Britain to the 8th century).

This paper will likely also have its controversies, but for now it is the most modern hypothesis that we have.

I hope this has explained what exactly is controversial about the “Anglo-Saxon” migrations into Britain (as even the name itself has controversy), and how the hypotheses have changed over time, from a large-scale invasion and slaughter to a small-scale elite migration to a large-scale migration over a long period of time. The scant sources we have on this period means that it will likely always be subjected to controversy, but archaeological, material, and DNA evidence has been able to illuminate this period to a certain degree.

I am not an expert on this period by any means, so if a more knowledgeable person finds gaps in my answer please do let me know so I can amend it.

Sources:

Gretzinger, J., Sayer, D., Justeau, P. et al. The Anglo-Saxon migration and the formation of the early English gene pool. Nature 610, 112–119 (2022). https://doi.org/10.1038/s41586-022-05247-2.

Thomas, M., Stumpf, M., Härke, H., Evidence for an apartheid-like social structure in early Anglo-Saxon England. (2006) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1635457/#!po=0.847458.

J.A. Giles’s translation of Gildas’s work: https://www.gutenberg.org/cache/epub/1949/pg1949.html (chapter 24).

edit: attempted to fix formatting