  We consider the quantum spin-$s$ $XXZ$ Heisenberg antiferromagnet on the two-
and three-dimensional pyrochlore lattices and examine a spin-Peierls mechanism
of lowering the total energy by a lattice distortion in a high magnetic field.
For the exact eigenstates consisting of several independent localized magnons
in a ferromagnetic environment we show the existence of a spin-Peierls
instability by rigorous analytical calculations. In addition we report on exact
diagonalization data for finite two-dimensional pyrochlore lattices up to N=64
sites. We discuss a scenario of the field-tuned changes in the ground-state
properties of the frustrated spin lattices due to the coupling between spin and
lattice degrees of freedom.
