currently /particle falling_leaves:falling_leaf doesnt have a parameter for the block being used
it could be useful to have a block id paramater there that changes the color of the particle just like minecraft:falling_dust

Gave this one a try and was able to make it work, have some lava leaves!

There's a big limitation to this however: This will only work in singleplayer.

That's because the command is executed server-side and since Falling Leaves is a client-side mod, the server doesn't know about the falling_leaf particle. There's no way around this except making Falling Leaves a server-side mod too, which simply does not feel right for such a small feature - I think such a command block would then also cause errors on vanilla clients.
So singleplayer is the best I can do here.