Phase I: 445-479
Main area: Kent
Intense fighting between 457 and 460
More fighting 470-471
Another battle 478
No further victories
No spread beyond Medway (origin of Men of Kent/Kentishmen split?)
Phase II: 483- ca 500
Main area: Sussex
Anderida sacked in 490s
No further victories
No spread beyond Weald to North
Phase III: 502 onwards
Main area: Wessex
Fighting in area of landings (two or three waves)
Nothing conclusive until 530, when Isle of Wight is taken
First real break-out in 554 (Sarum)
Note mixed Celtic and Saxon names of kings
In other words, the traditional picture of the Saxons flooding in and across the country isn't supported even by the ASC, certainly not for the South of England; and the archaeology tends to bear this out.
Phase I saw a lot of organised fighting, apparently by Ambrosius Aurelianus (the elder, by some reckoning) that took place in three bursts with intervals of 8-10 years between them. The 8 years figure may be an approximation due to the use of Easter Tables, but would not be wildly out, and tradition would retain the idea of longish spells between the fighting. It is also necessary to sort out exactly what was happening with Vortigern and Vortimer; there may well have been conflict between Vortigern and the more Romanised forces.
Phase II starts fairly quietly; possibly there were not many inhabitants on the Sussex coast to be bothered. It is only after about 10 years that the attack on Anderida comes, followed by other fighting, possibly climaxing at Badon. This might have involved Ambrosius Aurelianus (the younger - see above); it is tempting to consider the possibility that he was Count of the Saxon Shore and would have had definite responsibility.
There does not seem to be any evidence that they moved North of the Weald; all the activities seem to take place along the South Coast.
Phase III is a mess. The names are completely untrustworthy apart from Cerdic's immediate family - who have a surprising number of Celtic names; there are at least two waves of invasion, three if Cerdic came in from outside; there is vast uncertainty as to whether or not the leaders of the later waves are related to Cerdic. The only thing that is certain is that apart from a few inconclusive battles nothing really happened until the taking of the Isle of Wight (and some Welsh sources suggest this was partly by agreement anyway). There is no break out from the coastal area until Sarum is taken - in the second half of the 6th century.
OK, the dates are dodgy and at least some of the names are dodgy but the Anglo-Saxon Chronicle (as confirmed quite often by Nennius) is about all we have to go by for what was happening in South East England.
I'll come back later with possible interpretations of what was happening, and on what was happeing in the West.