How is cache coherency handled?

The hardware usually manages cache coherency and there are many different strategies to ensuring caches are consistent. If you're interested in this (warning: This is a deep rabbit hole), then I suggest this as a good starting point:

A Primer on Memory Consistency and Cache Coherence, Second Edition https://www.morganclaypool.com/doi/abs/10.2200/S00962ED2V01Y201910CAC049