I haven't read all of the research around Anglo-Saxons/Indigenous Brits in particular, but I can give you the context of the study of human migration via genetics that my paleoanthropology professors gave me. Generally how these studies are done are by measuring one of two things, either a degree of variance in particular parts of the DNA, or the presence or absence of particular genes or gene sets that are known to be a part of a particular population. Now, I'm not sure which was utilized in the studies that are used in the arguments of degree of gene flow from various migrations into Britain, but both methods are able to give insight only, they cannot be read as definitive proof of the presence or absence of invading populations for several reasons.1.) Mutations are recurring. That is to say that a genetic mutation, a variance in your DNA from others' DNA, is a built in probability. The same mutations can and will spontaneously show up in separate populations. No contact between populations may be needed. 2.) A period of gene flow (one population meeting another and combining) may be later masked by later gene flows, or simply disappear through time. Say the Anglo-Saxons did invade - and big time. But then they hung around by the coasts for a few generations, and no new Anglo-Saxons came in - over time, they're mixing with more and more Northern, non Anglo-Saxon people. The Anglo-Saxon genetic markers may well be swamped out and disappear. That doesn't mean the invasion didn't happen, just that the gene flow wasn't large or long enough to make a lasting genetic impact. 3.) Mutations at the population level take time. Lots of it. We're talking about a period of less than 2,000 years. Using solely genetics to examine this problem is putting a lot of analytic pressure on not very many mutations. It's a bit like trying to study the movement patterns of protozoa using the Hubble telescope. 4.) This science is in its infancy. For one, there is no significant genetic difference between a Swede, an Inuit, and an Australian Aborigine. We have nothing markedly different from each other on a genetic level. Trying to discern discrete genetic populations in humans is like trying to distinguish green from extra-green. We're still trying to figure out how to map out significant genetic markers between Europeans and Indonesians. The idea that you could reliably tell the difference between different ancient European groups is very likely premature. Genetically, there simply wasn't a whole lot of difference between Indigenous Brits and the Anglo-Saxons. And it's not like we can reliably say these two groups weren't already in some genetic contact before the invasion in question. I love studying human migration - and our DNA is a very helpful tool in mapping out our own history. That said, it is only one tool among many and any good theory will be upheld through the application of diverse methods. In this context, genetic studies should in no way be read as indisputable or definitive proof of anything, they are simply one piece to the puzzle.