On the totally basic principle core 1 will tell core 2 (say non-dirty state) to 'invalidate' it. At the same time core 1' and '2' can read a different value for that memory address/cache line before '2' carries the task. The protocol is more complex than that, the cache lines have different states. The basic protocol is known as MESIF[0] for Intel.

https://en.wikipedia.org/wiki/MESIF_protocol

Core1 would send a invalidation to get exclusive (write) access to that cache line. This would be routed to Core2. Which removes the cache line from its cache.

If core2 now accesses that cache line again, it would not find it in its L1, or L2, and at L3 there is the cacheline directory routing the request to core1.