  We consider the graphical mean curvature flow of maps ${\bf
f}:\mathbb{R}^m\to\mathbb{R}^n$, $m\ge 2$, and derive estimates on the growth
rates of the evolved graphs, based on a new version of the maximum principle
for properly immersed submanifolds that extends the well-known maximum
principle of Ecker and Huisken derived in their seminal paper [10]. In the case
of uniformly area decreasing maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^2$, $m\ge
2$, we use this maximum principle to show that the graphicality and the area
decreasing property are preserved. Moreover, if the initial graph is
asymptotically conical at infinity, we prove that the normalized mean curvature
flow smoothly converges to a self-expander.
