Metazoan development proceeds through a series of morphogenetic events that sculpt body plans and organ structures. In the early embryo, these processes occur concurrently such that forces generated in neighbouring tissues can impose mechanical stresses on each other, potentially disrupting development and consequently decreasing fitness. How organisms evolved mechanisms to mitigate inter-tissue mechanical conflicts remains unclear. Here, we combined phylogenetic survey, quantitative live imaging and functional mechanical perturbation to investigate mechanical stress management during gastrulation across the insect order of flies (Diptera). We identify two distinct cellular mechanisms that prevent tissue collision between the expanding head and trunk. In Cyclorrhapha, a monophyletic subgroup including Drosophila melanogaster, active out-of-plane deformation of a transient epithelial fold, called the cephalic furrow, acts as a mechanical sink to pre-empt head-trunk collision. Genetic and optogenetic ablation of the cephalic furrow leads to accumulation of compressive stress, tissue buckling at the head-trunk boundary and late-stage embryonic defects in the head and nervous system. By contrast, the non-cyclorrhaphan Chironomus riparius lacks cephalic furrow formation and instead undergoes widespread out-of-plane division that reduces the duration and spatial extent of head expansion. Re-orienting head mitosis from in-plane to out-of-plane in Drosophila partially suppresses tissue buckling, showing that it can function as an alternative mechanical sink. Our data suggest that mechanisms of mechanical stress management emerge and diverge in response to inter-tissue conflicts during early embryonic development.