Although I don’t believe in reincarnation, I’ll try to answer this.

Before I attempt to answer this, however, I should point out that different systems, which hold to reincarnation, may address your objection differently. Many Ancient Indian philosophers, for example, would debate the merits of an alternative school’s views on reincarnation (and karma) and attempt to unravel problems with them. The particular objection you raised is similar to one that’s already been addressed by various schools of thought, for quite a while. The objection (in my own words) is that the amount of living things seem to be increasing, so, if reincarnation is true, where are all of these souls coming from? The answer was to postulate different worlds that had life on them. Some of the souls from those worlds were being reincarnated here on Earth. There are an uncountable amount of such worlds, too. (They also higher realities, which may count as one of these worlds.)

So, to give a similar response to your objection (with some updated science), if everything went extinct on earth, then souls would simply be reincarnated on another planet. If everything went extinct in this universe, then souls would be reincarnated in another universe. There are an infinite amount of universes (or, at least, a lot of universes, if one finds the concept of mathematical infinities existing in reality metaphysically problematic (e.g. a Hilbert’s Hotel of universes)).

There are other ways to address this objection, too, but I think that’ll do it, for now.

I hope that helps. :-)