The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$, and I'm wondering how big a difference this is. (And under some additional conditions on a specific case I'm working on, the same but for 2-generated subgroups.)

Towards that end, let's say a (finite) group $G$ is $k$-locally nilpotent* of class $\leq c$ if every subgroup generated by $k$ elements has nilpotency class $\leq c$, and let us call the smallest such $c$ the $k$-local nilpotence class of $G$.

For finite $p$-groups of 2-local (resp. 3-local) nilpotence class $c$, is there an upper bound on the (global) nilpotency class? What are examples achieving the largest known gap between the 2-local (resp. 3-local) nilpotency class and the (global) nilpotency class?

I'd also be interested in the same question for $p$-Lie algebras.

I've found it quite hard to search for — there seems to be a lot of literature about groups satisfying Engel conditions, but these aren't quite the same. A group of 2-local nilpotence class $c$ is $c$-Engel, but I believe the converse isn't true in general. And there's a lot of literature about locally nilpotent groups, but that is usually in the setting of infinite groups (where every finitely generated subgroup is nilpotent, or even where every f.g. subgroup is finite nilpotent). And of course "2-local" and "3-local" have their own meanings in finite group theory, so the overlapping/non-standard nomenclature really gets in the way.

While of course an answer would be great, because of the difficulty of searching, I'd also be happy with better keywords, authors, pointers to the literature, etc.

* Maybe not the best name, but I don't know a better one - if you do please let me know in the comments!