Rigidity lemma up to cover Let $X,Y,Z$ be [Edit: normal] proper varieties over $\mathbb{C}$. Let $W\to X\times Y$ be a finite flat surjective morphism, and let $W\to Z$ be a morphism.
Fix $x \in X$ and suppose that $\{x\}\times Y$ is contracted to a point, say $z$, in $Z$ in the following sense:
The fibre over $\{x\}\times Y$ along $W\to X\times Y$ is contracted to $Z$ via $W\to Z$.
If  $W\to X\times Y$ is an isomorphism, then the Rigidity Lemma says that $W\cong X\times Y\to Z$ factors over $X\to Z$. (The map is "independent of $Y$" in some sense.)
Is there some version of the Rigidity Lemma that we can apply in the above situation to see that $W\to Z$ is also "independent of $Y$"? More precisely, I expect that there is a  finite flat morphism $W'\to W$ such that $W'\to Z$ factors over some covering of $X$.
 A: This is not an answer but really a long comment. Suppose also that all the varieties in sight are smooth.
Let $p:W\to X$ be the composition of the morphism
$W\to X\times Y$ with the projection on the first factor and let $g:W\to Z$ be the morphism you consider. Let $f:W\to X\times Z$ be the morphism
st $f(w)=(p(w),g(w))$. Let $q:X\times Z\to X$ be the
projection on the first factor. Then we have trivially
$q\circ f=p$ so that $f$ is an $X$-morphism.
If for all $x\in X$, we have $\dim_{\kappa(x)}(W_x,{\cal O}_{W_x})=1$, then you can apply Mumford's Prop. 6.1 in GIT to deduce what your want (by letting $X\to Z$ be $\eta$ composed with the projection on the second factor). In your situation, this last assumption will not be satisfied in general though. However, if the morphism
$W\to X\times Y$ is (finite and) étale then the morphism
$p$ has geometrically reduced fibres, is cohomologically flat in degree $0$ and has a Stein factorisation
$W\stackrel{p'}{\to} X'\to X$, where $p'$ satisfies the above
assumption and $X'\to X$ is 'etale and finite. So if $W\to X\times Y$ is 'etale,
there is a morphism $X'\to Z$ such that $g$ factors through
$p'$.
I am not sure that there is a similar factorisation in general (I believe one could construct a counterexample but I don't have one to hand).
