  We review the dynamics of interacting particles in disorder-free potentials
concentrating on a combination of a harmonic binding with a constant tilt. We
show that a simple picture of an effective local tilt describes a variety of
cases. Our examples include spinless fermions (as modeled by Heisenberg spin
chain in a magnetic field), spinful fermions as well as bosons that enjoy a
larger local on-site Hilbert space. We also discuss the domain-wall dynamics
that reveals nonergodic features even for a relatively weak tilt as suggested
by Doggen et. al. [arXiv:2012.13722]. By adding a harmonic potential on top of
the static field we confirm that the surprizing regular dynamics is not
entirely due to Hilbert space shuttering. It seems better explained by the
inhibited transport within the domains of identically oriented spins. Once the
spin-1/2 restrictions are lifted as, e.g., for bosons, the dynamics involve
stronger entanglement generation. Again for domain wall melting, the effect of
the harmonic potential is shown to lead mainly to an effective local tilt.
