  We calculate the rates of phonon-assisted hyperfine spin flips during
electron and hole tunneling between quantum dots in a self-assembled quantum
dot molecule. We show that the hyperfine process dominates over the
spin-orbit-induced spin relaxation in magnetic fields up to a few Tesla for
electrons, while for holes this cross-over takes place at field magnitudes of a
fraction of Tesla, upon the assumption of a large $d$-shell admixture to the
valence band state, resulting in a strong transverse hyperfine coupling. The
interplay of the two spin-flip mechanisms leads to a minimum of the spin-flip
probability, which is in principle experimentally measurable and can be used as
a test for the presence of substantial transverse hyperfine couplings in the
valence band.
