Vague question: Is there a criterion to deduce that a morphism between algebraic stacks is finite based on the local deformation functors?

For sure this is not enough, so let me be more specific.

Suppose that $f: X \to Y$ is a morphism of algebraic stacks. Let us assume in addition that the diagonal $\Delta_f : X \to X \times_Y X$ is finite and unramified.

Now assume that for each point $x \in |X|$ of finite type over $y \in |Y|$ the morphism between the (pro-representing rings of the) deformation functors is finite.

Specific question: Under these conditions, can we assume that $f \colon X \to Y$ is finite?

Edit: This too is not enough as Piotr demonstrates in the comments. The key observation is that finiteness is not local on the source and even an open immersion is a counterexample to the question above.

Let me then modify the question like so:

Modified question: What global criteria, such as properness, on the morphism $f:X \to Y$ need to be assumed to make finiteness (formal) local on the source?