  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.
