  We simulate the propagation of cosmic rays at ultra-high energies, $\gtrsim
10^{18}$ eV, in models of extragalactic magnetic fields in constrained
simulations of the local Universe. We use constrained initial conditions with
the cosmological magnetohydrodynamics code {\sc ENZO}. The resulting models of
the distribution of magnetic fields in the local Universe are used in the
\crpropa code to simulate the propagation of ultra-high energy cosmic rays. We
investigate the impact of six different magneto-genesis scenarios, both
primordial and astrophysical, on the propagation of cosmic rays over
cosmological distances. Moreover, we study the influence of different source
distributions around the Milky Way. Our study shows that different scenarios of
magneto-genesis do not have a large impact on the anisotropy measurements of
ultra-high energy cosmic rays. However, at high energies above the GZK-limit,
there is anisotropy caused by the distribution of nearby sources, independent
of the magnetic field model. This provides a chance to identify cosmic ray
sources with future full-sky measurements and high number statistics at the
highest energies. Finally, we compare our results to the dipole signal measured
by the Pierre Auger Observatory. All our source models and magnetic field
models could reproduce the observed dipole amplitude with a pure iron injection
composition. Our results indicate that the dipole is observed due to clustering
of secondary nuclei in direction of nearby sources of heavy nuclei. A light
injection composition is disfavoured by the non-observation of anisotropy at
energies of $4-8 \rm\ EeV$.
