Cross platform play

So Diablo 2 & 3 bundle are on sale for the switch. I dabbled in D3 via PC free trial on Blizzard.net account.

I hear D2 can be cross platform character progression. How does that work? Will I need to buy the game across each platform I play on? Is the character tied to my Blizzard account? And although this is a D2 thread, if you can answer for D3 too I would appreciate it. Thanks!