Let P be a set of polygonal pseudodiscs in the plane with n edges in total translating with fixed velocities in fixed directions. We prove that the maximum number of combinatorial changes in the union of P is \Theta(n 2 ff(n)). In general, if the pseudodiscs move along curved trajectories, then the maximum number of changes in the union is \Theta(n s+2 (n)), where s is the maximum number of times any triple of polygon edges meet in a common point. We apply this result in two different settings. First, we prove that the complexity of the free space of a constant-complexity polygon translating amidst convex polyhedral obstacles with n edges in total is O(n 2 ff(n)). Second, we show that the complexity of the space of lines missing a set of n convex homothetic polytopes of constant complexity in 3-space is O(n 2 4 (n)). Both bounds are almost tight in the worst case. 1 Introduction Let P be a set of polygons in the plane with n edges in total. Each polygon translates with a fixed ..