  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 due to
global conservation moments. It is rather strongly linked to 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.
