  In this article we study various forms of $\ell$-independence (including the
case $\ell=p$) for the cohomology and fundamental groups of varieties over
finite fields and equicharacteristic local fields. Our first result is a strong
form of $\ell$-independence for the unipotent fundamental group of smooth and
projective varieties over finite fields, by then proving a certain `spreading
out' result we are able to deduce a much weaker form of $\ell$-independence for
unipotent fundamental groups over equicharacteristic local fields, at least in
the semistable case. In a similar vein, we can also use this to deduce
$\ell$-independence results for the cohomology of semistable varieties from the
well-known results on $\ell$-independence for smooth and proper varieties over
finite fields. As another consequence of this `spreading out' result we are
able to deduce the existence of a Clemens--Schmid exact sequence for formal
semistable families. Finally, by deforming to characteristic $p$ we show a
similar weak version of $\ell$-independence for the unipotent fundamental group
of a semistable curve in mixed characteristic.
