The initial acceleration does not matter at all. You could completely leave it out by assuming an incoming human A on his spaceship at Vrel which just happens to be of the same age when he passes another human B sitting on earth.
A will see B's clock move slower, as in calculate it to move slower (time dilation)
B will see A's clock move slower, as in calculate it to move slower.
This is a symmetrical situation. It would have been the same if B was formerly at rest and then decided to instantaneously accelerate towards a direction away of A.
So who will be the younger one, even though both see(as in calculate) each other as aging slower.
It will be whoever of the two accelerates into the inertial frame of reference, the other is at rest in, in the second phase.
So if A accelerates into B's rest frame after some time, A will be younger one.
If B accelerates into A's rest frame after some time of moving apart, then B will be the younger one.
This is phase two.
In phase 3 one of them would decide to accelerate back and meet up again. Again, whoever decides to do that acceleration or to state it more precisely, whoever decides to switch into a different rest frame, will end up being younger.
But it's not the acceleration which does the trick. If it was the acceleration, accelerating instantaneously into another frame, and then back again instantaneously, you should end up younger. Which is not the case, because locally your clocks remain in sync with whichever clock you left in the former rest frame.
It's the switching of your rest frame which is what causes the age difference.
Let's check phase 3 isolated.
B is at a distance to A. Assume they are both of the same age.
There is a long line of clocks going out from B to A. All clocks will be synced, so B's clock, A's clock and all other clocks along the path now display zero. B plots this scenario on a x,t diagram.
B instantaneously accelerates to Vrel. He is now in a different rest frame.
He can plot another x,t diagram and calculate what each clock along the path to A will display along the x axis (at the same time, in the classical sense)
He will soon find out, that none of the clocks are in sync anymore. His clock displays zero still, but clocks towards A have a higher count with A's clock at the end having the highest count.
That's what relativity of simultaneity is about. No two events which are space separated and happen at the same time in the classical sense (events that are on a line parallel to the x axis - therefore, having the same t value) do happen at the same time seen from an observer in a different rest frame.
And even though B will see A's clock (as in calculate using SR) to be ticking slower than his clock, it won't be enough to make up for the time shift of A's clock, he calculated, when switching his rest frame, until he reaches A.
A on the other hand did not switch his rest frame. All he calculates while B is approaching, is B's clock ticking slower. The acceleration of B at a distance had no effect on A other than just B now approaching at Vrel towards him.
And so B will end up to be the younger one when they meet.
So SR can fully explain this scenario, but it's not a really good explanation, allowing you to fully understand what is going on.
One has to understand what switching of a rest frame means geometrically and for that he would have to be able to imagine spacetime as a hypercube.
Because this switching of the rest frame is merely a perspective change on this 4 dimensional construct.
In reality there is no motion at all. Spacetime is absolute with us looking at different slices from different perspectives.
If you were able to map all events that happened and will ever happen within this hypercube, there won't be anything moving. It's static/absolute with all events causally linked. We are all looking at the same hypercube, no matter which rest frame we are in. It just looks different to us depending on the perspective which is what rest frames are in essence. A perspective change.
Acceleration causes this perspective change, but we do not really know how exactly it does that. We explain it with motion caused by electrons repelling each other and so on, but i don't think that goes deep enough.