  We derive Onsager-Machlup functionals for countable product measures on
weighted $\ell^p$ subspaces of the sequence space $\mathbb{R}^{\mathbb{N}}$.
Each measure in the product is a shifted and scaled copy of a reference
probability measure on $\mathbb{R}$ that admits a sufficiently regular Lebesgue
density. We study the equicoercivity and $\Gamma$-convergence of sequences of
Onsager-Machlup functionals associated to convergent sequences of measures
within this class. We use these results to establish analogous results for
probability measures on separable Banach or Hilbert spaces, including Gaussian,
Cauchy, and Besov measures with summability parameter $1 \leq p \leq 2$.
Together with Part I of this paper, this provides a basis for analysis of the
convergence of maximum a posteriori estimators in Bayesian inverse problems and
most likely paths in transition path theory.
