Sufficient and necessary conditions are established for controllability of affine control systems, motivated by these with a drift all of whose solutions are periodic. In contrast with previously known results, these conditions encompass the case of a control set whose convex hull contains the origin but is not a neighborhood of the origin. The condition is expressed by means of pushforwards along the flow of the drift, rather than in terms of Lie brackets. It turns out that this also amounts to local controllability of a time-varying linear approximation with constrained controls. Global and local results are given, as well as a few illustrative examples.