in some of the worlds you are spirited away at near light-speed by evil aliens, and the measure of those worlds becomes relatively large once your "stationary" (in some sense) copies die out.
What matters, presumably, is the relative measure of worlds within the "branch" that you're "already in". Or, in other words, once you're in branch a_1 (rather than b_1) at time t_1, the probability that you end up in branch a_{12} at time t2 is the conditional probability P(a_{12}|a_1) rather than the prior P(a_{12}) at time t_0. So it's very unlikely to jump into a b-branch future from an a-branch past.
There is a subset of worlds in the branch "a" (or any other branch) where the aliens carry away your body at high speed. And this branch will eventually dominate any other subbranches (where you are still alive) of the same branch.
[Today, I participated in a conversation about the idea of quantum immortality. I decided to summarise some of the thoughts that came up in this short post. Therefore, it should be viewed as a report on a discussion rather than an attempt at a proper post.]