  Cosmic rays (CRs) have critical impacts in the multiphase interstellar medium
(ISM), driving dynamical motions in low-density plasma and modifying the
ionization state, temperature, and chemical composition of higher-density
atomic and molecular gas. We present a study of CR propagation between the
ionized ISM and a neutral cloud. Using one-dimensional magnetohydrodynamic
particle-in-cell simulations which include ion-neutral drag to damp
Alfv$\acute{\text{e}}$n waves in the cloud, we self-consistently evolve the
kinetic physics of CRs and fluid dynamics of the multiphase gas. By introducing
the cloud in our periodic domain, our simulations break translational symmetry
and allow the emergence of spatial structure in the CR distribution function. A
negative spatial gradient forms across the fully-ionized ISM region while a
positive gradient forms across the neutral cloud. We connect our results with
CR hydrodynamics formulations by computing the wave-particle scattering rates
as predicted by quasilinear, fluid, and Fokker-Planck theory. For momenta where
the mean free path is short relative to the box size, we find excellent
agreement among all scattering rates. By exploring different cloud sizes and
ion-neutral collision rates, we show that our results are robust. Our work
provides a first-principles verification of CR hydrodynamics when particles
stream down their pressure gradient, and opens a pathway toward comprehensive
calibrations of transport coefficients from self-generated
Alfv$\acute{\text{e}}$n wave scattering with CRs.
