  It has been proposed that Ultra-High-Energy Cosmic Rays (UHECRs) up to
$10^{20}$eV could be produced in the relativistic jets of powerful Active
Galactic Nuclei (AGNs) via a one-shot reacceleration of lower-energy CRs
produced in supernova remnants (the $\textit{espresso}$ mechanism). We test
this theory by propagating particles in realistic 3D magneto-hydrodynamic
simulations of ultra-relativistic jets and find that about $10\%$ of the CRs
entering the jet are boosted by at least a factor of $\sim\Gamma^2$ in energy,
where $\Gamma$ is the jet's effective Lorentz factor, in agreement with the
analytical expectations. Furthermore, about $0.1\%$ of the CRs undergo two or
more shots and achieve boosts well in excess of $\Gamma^2$. Particles are
typically accelerated up to the Hillas limit, suggesting that the
$\textit{espresso}$ mechanism may promote galactic-like CRs to UHECRs even in
AGN jets with moderate Lorentz factors, and not in powerful blazars only.
Finally, we find that the sign of the toroidal magnetic field in the jet and in
the cocoon controls the angular distribution of the reaccelerated particles,
leading to a UHECR emission that may be either quasi-isotropic or beamed along
the jet axis. These findings strongly support the idea that $\textit{espresso}$
acceleration in AGN jets can account for the UHECR spectra, chemical
composition, and arrival directions measured by Auger and Telescope Array.
