  This article concludes a series of papers (R. Folk, Yu. Holovatch, and G.
Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009))
where the tools of the field theoretical renormalization group were employed to
explain and quantitatively describe different types of static and dynamic
behavior in the vicinity of multicritical points. Here, we give the complete
two loop calculation and analysis of the dynamic renormalization-group flow
equations at the multicritical point in anisotropic antiferromagnets in an
external magnetic field. We find that the time scales of the order parameters
characterizing the parallel and perpendicular ordering with respect to the
external field scale in the same way. This holds independent whether the
Heisenberg fixed point or the biconical fixed point in statics is the stable
one. The non-asymptotic analysis of the dynamic flow equations shows that due
to cancelation effects the critical behavior is described - in distances from
the critical point accessible to experiments - by the critical behavior
qualitatively found in one loop order. Although one may conclude from the
effective dynamic exponents (taking almost their one loop values) that weak
scaling for the order parameter components is valid, the flow of the time scale
ratios is quite different and they do not reach their asymptotic values.
