Is there an analogue of Knudsen clutching for moduli stacks of "pointed" varieties?

I admit this question is not very precise. I'm really asking two questions though.

Are there analogues of the moduli stack of (g,n) curves in higher dimension, where we consider say varieties with hilbert polynomial h and n "points" where I don't know yet what I mean by points precisely?

Do these moduli stacks, if they exist at all, have some kind of Knudsen clutching?

Still not precise, but hopefully a bit more clear.