  Entanglement swapping and heralding are at the heart of many protocols for
distributed quantum information. For photons, this typically involves Bell
state measurements based on two-photon interference effects. In this context,
hybrid systems that combine high rate, ultra-stable and pure quantum sources
with long-lived quantum memories are particularly interesting. Here, we develop
a theoretical description of pulsed two-photon interference of photons from
dissimilar sources to predict the outcomes of second-order cross-correlation
measurements. These are directly related to, and hence used to quantify, photon
indistinguishability. We study their dependence on critical system parameters
such as quantum state lifetime and frequency detuning, and quantify the impact
of emission time jitter, pure dephasing and spectral wandering. Our results
show that for fixed lifetime of emitter one, for each frequency detuning there
is an optimal lifetime of emitter two that leads to highest photon
indistinguishability. Expectations for different hybrid combinations involving
III-V quantum dots, color centers in diamond, 2D materials and atoms are
quantitatively compared for real-world system parameters. Our work both
provides a theoretical basis for the treatment of dissimilar emitters and
enables assessment of which imperfections can be tolerated in hybrid photonic
quantum networks.
