Nonequilibrium energy transfer can drive nanoscale quantum systems into excited states and generate dynamics that are absent in equilibrium. Here, we report real-time measurements of phonon-driven charge and spin dynamics at the triplet resonance of a semiconductor double quantum dot. Nonequilibrium acoustic phonons emitted by a nearby biased quantum dot induce resonant interdot tunneling between the (0,2) and (1,1) triplet states. Time-resolved charge sensing reveals stochastic tunneling dynamics governed by Pauli spin blockade and the competition between interdot and dot–reservoir tunneling processes. When the interdot tunneling rate exceeds the dot–reservoir tunneling rates, repeated triplet-mediated interdot transitions are observed. In the weaker interdot-coupling regime, where the interdot and dot–reservoir tunneling rates become comparable, the nonequilibrium phonon environment instead drives a unidirectional electron-transport cycle along the phonon-density gradient. These results demonstrate how spin blockade and competing tunneling processes govern phonon-driven nonequilibrium dynamics and directional electron transport in a multilevel quantum-dot system.