  In 2010, Hrushovski--Loeser showed that the Berkovich analytification of a
quasi-projective variety over a non-Archimedean valued field admits a
deformation retraction onto a finite simplicial complex. In this article, we
adapt the tools and methods developed by Hrushovski--Loeser to study if such
deformation retractions can be obtained to be compatible with respect to a
given morphism. Amongst other results, we show that compatible deformation
retractions exist over a constructible partition of the base and prove the
general statement in the case of a morphism of relative dimension 1 where the
target is a smooth connected curve.
