How does an ālaya-vijñāna work? And in particular, is it eternal, unchanging and like any other formulation for a soul that we might have seen, in say Christianity or Hinduism?

This particular idea, that ālaya-vijñāna is like a permanent self, is mentioned in "Confessions of a Buddhist Atheist", by Stephen Bachelor.
In no way alaya-vijnana is "eternal, unchanging". This is Buddhism after all and nobody will posit totally non-buddhistic concepts which could contradict basic teachings of Buddha. Unenlightened beings may mistakenly perceive alaya-vijnana as Soul, but they may perceive so about any skandha. (One interpretation of why skandhas are five is that it's five assumptions of Self.) Standard analogy for alaya-vijnana is violent current of a river (udakaugha), it may be perceived as solid (from a far) but it's highly changing (if looked closer). Alaya-vijnana is only remains to the end of sansara.