  We prove a finiteness theorem and a comparison theorem in the theory of
\'etale cohomology of rigid analytic varieties. By a result of Huber, for a
quasi-compact separated morphism of rigid analytic varieties with target being
of dimension $\le1$, the compactly supported higher direct image preserves
quasi-constructibility. Though the analogous statement for morphisms with
higher dimensional target fails in general, we prove that, in the algebraizable
case, it holds after replacing the target with a modification. We deduce it
from a known finiteness result in the theory of nearby cycles over general
bases and a new comparison result, which gives an identification of the
compactly supported higher direct image sheaves, up to modification of the
target, in terms of nearby cycles over general bases.
