We investigated the dynamically modified interaction between Rydberg atoms and a single mode of a microwave cavity driven by a strong external field. With a monochromatic injected field we observed dynamic suppression of spontaneous emission, when the intracavity Rabi frequency exceeds the cavity linewidth. We discuss a theoretical model using non-Markovian decay rates, which is in excellent agreement with experiment. When a bichromatic driving field is used, enhanced decay reappears in narrow resonance zones as the detunings and injected intensities are varied. These results are interpreted in terms of cavity-modified transitions between the Floquet states of a two-level atom. Striking aspects of these transitions are their small linewidth and an asymmetry between emission and absorption.