  Random walk on changing graphs is considered. For sequences of finite graphs
increasing monotonically towards a limiting infinite graph, we establish
transition probability upper bounds. It yields sufficient transience criteria
for simple random walk on slowly growing graphs, upon knowing the volume and
Cheeger constant of each graph. For much more specialized cases, we establish
matching lower bounds, and deduce sufficient (weak) recurrence criteria. We
also address recurrence directly in relation to a universality conjecture of
[DHS]. We answer a related question of [SZ, Problem 1.8] about "inhomogeneous
merging" in the negative.
