This paper examines leadership in internationalization strategies for an asymmetric cost duopoly where firms choose between exports and foreign direct investment (FDI) in a sequential setting. The incentive to lead and to engage in FDI is stronger for the more efficient firm. With sequential choices and the efficient firm playing in advance, it is less likely that firms pick identical internationalization strategies in equilibrium, as compared with simultaneous choices; this is more so for greater cost asymmetry. It also happens for large enough oligopoly profitability when the inefficient firm plays in advance. Follow-the-leader behaviour in FDI arises for low values of the setup cost. Although entry in FDI by both firms is best for consumers, total welfare can be higher with opposite internationalization strategies. Were firms given the opportunity to lead or wait and enter later, the efficient firm would emerge as the leader in exports/FDI depending on the well-known proximity-concentration trade-off. Interestingly, the less efficient firm might prefer to wait for strategic reasons.