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We develop a framework to systematically investigate the influence of many-particle interference on the dynamics of generic—possibly interacting—bosonic systems. We consider mixtures of bosons which belong to several distinguishable species, allowing us to tune the level of many-particle interference, and identify the corresponding signatures in the time-dependent expectation values of observables. Interference contributions to these expectation values can be classified based on the number of interfering particles. Interactions are shown to generate a series of additional, higher-order interference contributions. Finally, based on a decomposition of the Hilbert space of partially distinguishable bosons into irreducible representations of the unitary group, we determine some spectral characteristics of (in)distinguishability.