  The unconventional critical behavior near magnetic quantum phase transitions
in various heavy-fermion metals, apparently inconsistent with the standard
spin-density-wave scenario, has triggered proposals on the breakdown of the
Kondo effect at the critical point. Here we investigate, within one specific
scenario, the fate of such a zero-temperature transition upon coupling of the
electronic to lattice degrees of freedom. We study a Kondo-Heisenberg model
with volume-dependent Kondo coupling -- this model displays both Kondo volume
collapse and Kondo-breakdown transitions, as well as Lifshitz transitions
associated with a change of the Fermi-surface topology. Within a large-N
treatment, we find that the Lifshitz transition tends to merge with the Kondo
volume collapse and hence becomes first order, whereas the Kondo breakdown
transition remains of second order except for very soft lattices. Interesting
physics emerges at the zero-temperature endpoint of the Kondo volume collapse:
For electrons in two space dimensions, this endpoint is located at the Lifshitz
line for a large range of parameters, thus two continuous quantum phase
transitions coincide without fine tuning. We analyze the effective Landau
theory for such a situation and discuss critical exponents. Finally, we relate
our findings to current heavy-fermion experiments.
