  We study the relaxational critical dynamics of the three-dimensional random
anisotropy magnets with the non-conserved n-component order parameter coupled
to a conserved scalar density. In the random anisotropy magnets the structural
disorder is present in a form of local quenched anisotropy axes of random
orientation. When the anisotropy axes are randomly distributed along the edges
of the n-dimensional hypercube, asymptotical dynamical critical properties
coincide with those of the random-site Ising model. However structural disorder
gives rise to considerable effects for non-asymptotic critical dynamics. We
investigate this phenomenon by a field-theoretical renormalization group
analysis in the two-loop order. We study critical slowing down and obtain
quantitative estimates for the effective and asymptotic critical exponents of
the order parameter and scalar density. The results predict complex scenarios
for the effective critical exponent approaching an asymptotic regime.
