Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean
space when it develops a singularity of type I, and proved that its rescaled
flow converges to a self-shrinker in the Euclidean space. In this paper, we
generalize this result for a Ricci-mean curvature flow moving along a Ricci
flow constructed from a gradient shrinking Ricci soliton.