We use holography to analyze relativistic collisions in a one-parameter family of strongly coupled gauge theories with thermal phase transitions. For a critical value of the parameter, the transition is second order, for subcritical values it is first order, and for supercritical values it a smooth crossover. We extract the gauge theory stress tensor from collisions of gravitational shock waves on the dual geometries. Regardless of the nature of the transition, for values of the parameter close to the critical value, almost all the energy of the projectiles is deposited into a long-lived, quasistatic blob of energy at midrapidity. This configuration is well described by the constitutive relations of second-order hydrodynamics that include all second-order gradients that are purely spatial in the local rest frame. In contrast, a Müller-Israel-Stewart-type formulation of hydrodynamics fails to provide a good description. We discuss possible implications for searches of the QCD critical point.