  We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times
\Sigma_2, g_1-g_2)$ of a map $f:\Sigma_1\to \Sigma_2$ from a closed Riemannian
manifold $(\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold
$(\Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with
sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph,
exists for all time, and converges to a slice at infinity. We also show, with
no need of the assumption $K_2\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and
$K_2\leq -c$, $c>0$ constant, any map $f:\Sigma_1\to \Sigma_2$ is trivially
homotopic provided $f^*g_2<\rho g_1$ where
$\rho=\min_{\Sigma_1}K_1/\sup_{\Sigma_2}K_2^+\geq 0$, in case $K_1>0$, and
$\rho=+\infty$ in case $K_2\leq 0$. This largely extends some known results for
$K_i$ constant and $\Sigma_2$ compact, obtained using the Riemannian structure
of $\Sigma_1\times \Sigma_2$, and also shows how regularity theory on the mean
curvature flow is simpler and more natural in pseudo-Riemannian setting then in
the Riemannian one.
