It’d be nice to play the DLC classes with people who Don’t have the DLC. (Borderlands 2 style for example)
Welcome to the forum. 
It’s already possible to play with people who don’t have the expansions.
There’s no way you could play a DLC class in just the base game because it simply wouldn’t exist there. Those classes aren’t in the base game.
Unlikely if not impossible to do.
Imagine 2 players A and B in a Grim Dawn multiplayer game together. A has the expansions and B does not. B would need the expansion game files to see the masteries, skills and effects, items etc. that A has access to in the expansions. As the game is not server-based (unlike Borderlands 2), B would also have to have the game files locally to crossplay with A with no errors. This is also compounded by the fact that Grim Dawn heavily supports modding which again means the game files need to be freely made available to the player base.
This is something they cannot do as the expansions are paid content. At the end of the day, Crate is a business and they need to make revenue in some way to continue developing content and games.
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@Evil_Baka - Well said and to the point. I’ll probably bookmark this write-up to copy and paste for use on the Steam forum. We get so many there that have trouble wrapping their heads around why GD works the way it does.