Linguistic research shows that Old Frisian and Old English have a common ancestor, separate from Old Saxon.
In the 4th century AD the Dutch coast got depopulated and repopulated by Angles who became new Frisians, at the same time the Anglo-Saxon culture - as defined by material culture - appears in Central England, and Schleswig in Southern Denmark gets depopulated (all based on archaeological evidence).
These new Frisians kept close trade relations with England - around 600 AD they dominate trade around the North Sea - and kept a cultural split with the Saxons on the sandy high-grounds behind the coast, although in Groningen a strong cultural (pottery) influence between Sxons and Frisians has been observed, no surprise as Groningen forms one of the few gateways through the peat fields between the coast and the inland.
There were several settlements here that were continuously inhabited from 500 BC until present where one can observe a gradual change from old Chauci style to new Anglo-Saxon style in the same households.
Chauci are documented to have lived on artificial mounds at the coast as well as on the sandy hills in the inland by Roman writers, and their name is related to the word 'high'. They also appear as sea raiders on the Flemish coast around 200 AD, ad similar raiders a century later are called Saxons by the Romans.
But more to the West there was a real discontinuity with an 'empty' phase.
DNA research also shows that Central-English, Coastal Dutch and Southern Danish populations are very similar.
The Frisian area extended towards the Flemish coast in the Early Middle Ages, but got Frankified from the South, with Holland changing its language from Frisian to Frankish (Dutch) after 1000 AD.
So the picture on the continent is pretty clear wrt this 5th century migration, but I still wonder what happened to the previous British population in Central England?