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We theoretically and experimentally investigate the formation of dissipative coherent structures in Kerr nonlinear optical microresonators, whose integrated dispersion exceeds the free-spectral range. We demonstrate that the presence of any periodic modulation along the resonator's circumference, such as periodically varying dispersion, can excite higher-order comb structures. We explore the outcomes of coherent microcomb generation in both cases of anomalous and normal dispersion. We are able to access this regime in microresonators via the high peak power of synchronous pulse-driving. For 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. In normal dispersion, we see the coexistence of switching wave fronts with Faraday instability-induced period-doubling patterns, manifesting as powerful satellite microcombs highly separated either side of the core microcomb while sharing the same repetition rate. This regime of dispersion-modulated phase matching opens a dimension of Kerr cavity physics and microcomb generation, particularly for the spectral extension and tuneability of microcombs in normal dispersion.