Let N be a complete manifold with bounded geometry, such that secN≤−σ<0 for some positive constant σ. We investigate the mean curvature flow of the graphs of smooth length-decreasing maps f:Rm→N. In this case, the solution exists for all times and the evolving submanifold stays the graph of a length-decreasing map ft. We further prove uniform decay estimates for all derivatives of order ≥2 of ft along the flow.