Random searching for a mobile target is frequently encountered in many real situations, posing great challenges for theoretical analysis which traditionally deals only with static targets. We investigate mobile-object search on networks in which the target's location is changing with time. We adopt mean first-encounter time to quantify the search time a searcher takes to capture a time-moving target and derive its analytical expression, when it exists. Interestingly, we observe an entirely distinct behavior for a mobile-object search compared to traditional results with a static target. Counter-intuitively, we find that compared with searching for a static target, a mobile object is easier to be captured under the same circumstances. Furthermore, we demonstrate that staying at the hub node is the optimal strategy for hunting for a mobile object on a heterogeneous network. Our findings reveal distinct effects for a mobile object on both search and transport.