We study when worker and firm fixed-effects regressions describe wage changes associated with mobility and recover average causal effects of working at different firms. In a two-period setting, we show that, after adjusting for stayer wage growth, each firm contrast is a positively weighted average of signed difference-in-differences sums along paths in the mobility network. These adjusted firm contrasts reproduce the difference-in-differences contrasts for all observed types of moves if and only if the path sums are independent of the route taken, or equivalently sum to zero around every cycle. Conventional diagnostic plots can conceal violations of these restrictions. Under common trends, path independence also characterizes recovery of average effects for the workers making each move. For all supported moves, recovery of population average effects that is robust to nearby reweightings of movers and stayers additionally requires each move's mean gain to be equal across observed origin--destination groups. Applying the decomposition to Italian employer--employee data, we find that indirect paths account for most of the weight, on average, in our sampled comparisons between directly connected firms. (Joint work with Huang, Mogstad, Shaikh and Wilhelm)