This paper develops an idea flows theory of trade and growth with heterogeneous firms. New firms learn from incumbent firms, but the diffusion technology ensures entrants learn not only from frontier technologies, but from the entire technology distribution. By shifting the productivity distribution upwards, selection on productivity causes technology diffusion and this complementarity generates endogenous growth without scale effects. On the balanced growth path, the productivity distribution is a traveling wave with an increasing lower bound. Growth of the lower bound causes dynamic selection. Free entry mandates that trade liberalization increases the rates of technology diffusion and dynamic selection to offset the profits from new export opportunities. Consequently, trade integration raises long-run growth. The dynamic selection effect is a new source of gains from trade not found when firms are homogeneous. Calibrating the model implies that dynamic selection approximat ely triples the gains from trade relative to heterogeneous firm economies with static steady states.