Let $f:\mathcal{X}\rightarrow \mathcal{Y}$ be a morphism of
flows, $y$ an almost periodic point of $\mathcal{Y}$, and
$x\in f^{-1}(y)$. In general $x$ is not ne\-cessa\-rily almost
periodic, but several conditions are known under which that
happens. They fall into either ``compact" or ``noncompact"
conditions, depending on whether $\mathcal{X}$ and
$\mathcal{Y}$ are assumed to be compact or not. In
``noncompact" conditions other assumptions are restrictive. We
find a criterion for almost periodicity of $x$, which
generalizes both ``compact" and ``noncompact" statements at
the same time. We deduce theorems of Ellis, Markley,
Kutaibi-Rhodes and Pestov as corollaries.