  We present a formalism to study molecular polaritons in the collective
regime, based on the bosonization of molecular vibronic states. This formalism
accommodates an arbitrary number of molecules $N$, excitations and internal
vibronic structures, making it ideal for investigating molecular polariton
processes beyond the $N\rightarrow\infty$ limit (i.e., accounting for finite
$N$ effects), both numerically and analytically. Here, we employ this formalism
to rigorously derive radiative pumping and vibrational relaxation rates, and
establish the fundamental differences between these mechanisms. We show that
radiative pumping is the emission from incoherent excitons and divide its rate
into transmitted and re-absorbed components. On the other hand, the vibrational
relaxation rate in the weak linear vibronic coupling regime is composed of two
components: a $\mathcal{O}(1/N)$ contribution already accounted for by
radiative pumping, and a $\mathcal{O}(1/N^2)$ contribution from a second-order
process in the \textit{single}-molecule light-matter coupling that we identify
as a polariton-assisted Raman scattering. We demonstrate that vibrational
relaxation via this scattering is enhanced when the energy difference between
the emission and the lower polariton transition corresponds to the vibrational
excitation created in the Raman process.
