We study online optimization in a setting where an online learner seeks to optimize a per-round hitting cost, which may be non-convex, while incurring a movement cost when changing actions between rounds. We ask:under what general conditions is it possible for an online learner to leverage predictions of future cost functions in order to achieve near-optimal costs? Prior work has provided near-optimal online algorithms for specific combinations of assumptions about hitting and switching costs, but no general results are known. In this work, we give two general sufficient conditions that specify a relationship between the hitting and movement costs which guarantees that a new algorithm, Synchronized Fixed Horizon Control (SFHC), achieves a 1+O(1/w) competitive ratio, where w is the number of predictions available to the learner. Our conditions do not require the cost functions to be convex, and we also derive competitive ratio results for non-convex hitting and movement costs. Our results provide the first constant, dimension-free competitive ratio for online non-convex optimization with movement costs. We also give an example of a natural problem, Convex Body Chasing (CBC), where the sufficient conditions are not satisfied and prove that no online algorithm can have a competitive ratio that converges to 1.