Landau-Zener-St\“uckelberg interference – can occur. The types of modulation studied so far correspond to a continuous evolution of the system along the dispersion curve. Here we introduce a type of modulation called periodic latching, in which the qubit’s free phase evolution is interrupted by sudden switches in the transition frequency. In this case, the conventional Landau-Zener-St\“uckelberg theory becomes inadequate and we develop a novel adiabatic-impulse model for the evolution of the system. We derive the resonance conditions and we identify two regimes: a slow-modulation regime and a fast-modulation regime, in which case the rotating wave approximation (RWA) can be applied to obtain analytical results. The adiabatic-impulse model and the RWA results are compared with those of a full numerical simulation. These theoretical predictions are tested in an experimental setup consisting of a transmon whose flux bias is modulated with a square wave form. A rich spectrum with distinctive features in the slow-modulation and fast-modulation (RWA) regimes is observed and shown to be in very good agreement with the theoretical models. Also, differences with respect to the well known case of sinusoidal modulation are discussed, both theoretically and experimentally.