We theoretically and experimentally investigate the formation of dissipative coherent structures in Kerr nonlinear optical microresonators, whose spectrum encompasses an integrated dispersion that exceeds the cavity free-spectral range. We are able to access this regime in low-dispersion photonic chip-based microresonators by employing synchronous pulse driving, which increases the peak power over the continuous-wave-driving regime. Exploring this dispersion-folded regime, we demonstrate that the presence of periodically varying dispersion can excite higher-order comb structures, which we explore in both the normal and anomalous dispersion regimes. In the former, we observe the coexistence of switching wave fronts with Faraday instability-induced period-doubling patterns. They manifest as strong satellite microcombs highly separated either side of the core microcomb, at an offset close to half the repetition rate but while sharing the same repetition rate. In the latter, for dissipative Kerr solitons in anomalous dispersion, we observe the formation of higher-order phase-matched dispersive waves ("Kelly -like" sidebands), where the folded dispersion crosses the frequency comb grid. We observe up to the fifth higher-order dispersive wave in our experiments and show that these higher-order dispersive waves coherently extend the soliton frequency comb bandwidth significantly. For both cases, we show that our results can be understood by considering four-wave mixing in a two-dimensional Fourier transform representation. The results demonstrate the rich novel nonlinear dynamics of driven dissipative nonlinear cavities with periodically varying dispersion in the dispersion-folded pumping regime.