  Nearby radio galaxies (RGs) of Fanaroff-Riley Class I (FR-I) are considered
possible sites for the production of observed ultra-high-energy cosmic rays
(UHECRs). Among those, some exhibit blazar-like inner jets, while others
display plume-like structures. We reproduce the flow dynamics of FR-I jets
using relativistic hydrodynamic simulations. Subsequently, we track the
transport and energization of cosmic ray (CR) particles within the simulated
jet flows using Monte Carlo simulations. The key determinant of flow dynamics
is the mean Lorentz factor of the jet-spine flow,
$\langle\Gamma\rangle_{\rm{spine}}$. When
$\langle\Gamma\rangle_{\rm{spine}}\gtrsim$ several, the jet spine remains
almost unimpeded, but for $\langle\Gamma\rangle_{\rm{spine}}\lesssim$ a few,
substantial jet deceleration occurs. CRs gain energy mainly through diffusive
shock acceleration for $E\lesssim1$~EeV and shear acceleration for
$E\gtrsim1$~EeV. The time-asymptotic energy spectrum of CRs escaping from the
jet can be modeled by a double power law, transitioning from $\sim E^{-0.6}$ to
$\sim E^{-2.6}$ around a break energy, $E_{\rm{break}}$, with an exponential
cutoff at $E_{\rm{break}}\langle\Gamma\rangle_{\rm{spine}}^2$. $E_{\rm{break}}$
is limited either by the Hillas confinement condition or by particle escape
from the cocoon via fast spatial diffusion. The spectral slopes primarily arise
from multiple episodes of shock and relativistic shear accelerations, and the
confinement-escape processes within the cocoon. The exponential cutoff is
determined by non-gradual shear acceleration that boosts the energy of
high-energy CRs by a factor of $\sim \langle\Gamma\rangle_{\rm{spine}}^2$. We
suggest that the model spectrum derived in this work could be employed to
investigate the contribution of RGs to the observed population of UHECRs.
