If your thousands of instances are linked in a line, the youngest and eldest are affected only very slowly by blending (if the middle is not to be hugely overrepresented); that is why I came up with daily sex-switching. The deletion of one avatar each night is also implied by the merging.
But if you start with two copies of the chain, linked at matching ends to make a loop, then no deletion is necessary and the degree of blending can be the same all around; each-you imports yesterday’s memory from your clockwise and counterclockwise neighbors alternately, with a weight of 1/3 or less. A permanent body has social advantages, so I drop the automatic switching.
If a body dies, voluntarily or otherwise, an irregularity in the blending flows out, becoming smaller at each step. Why would you kill one of your bodies voluntarily? For one thing, I mean to keep the idea of alternating sexes around the loop, and you might find living as the wrong sex, even half the (subjective) time, intolerable. (I believe I am otherwise; else would I have the idea?)
If the friendly(?) aliens’ goal is to merge all of humanity into one meta-mind (think Childhood’s End), your two strands are joined only at the elder end; your baby ends are linked instead to other strands, typically of very distant cultures.
What if your private worldlet has no adults because you died as a minor? Some of everyone’s instances – would every third be enough? – ought to live in a shared world, each in a different town (though all near each other).
2022 Nov 04: I made a more stable page collecting my thoughts from this thread.
Since I gather you enjoy fantasy & SF, you might try in this context the ‘Chrestomanci’ novels by the late Diana Wynne Jones, starting with ‘Charmed Life’. Enchanters, it appears, have nine lives, at the same time — each in a different world/universe.