Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper?

I am interested in particular in the case when $X$ and $Y$ are both complete metric spaces.

Any help will be very much appreciated!

EDIT: Just to be clear, by *proper map* I mean a map such that the preimage of a compact set is a compact set.