This paper studies the responses of multi-destination exporters to import cost shocks in a context of variable markups. We develop a trade model with variable markups and we propose an empirical strategy that let us identify the within-firm, across-destinations elasticity of markup and the sensitivity of this elasticity to a firm's market power in the destination. On the empirical side, the methodology requires analyzing changes on firms' export values across destinations in response to exogenous cost shocks. We use a comprehensive data of Argentinian firms and exploit variability in the timing of import barriers imposed to Argentinian products. Not surprisingly, we find that trade barriers reduce imports for those firms that are more exposed to the policy. This, in turn, yields to a considerably decline in their total exports. According to our estimates, the elasticity of total exports with respect to total imports is around 50%. We then use the cost shock to uncover the main fact of this paper: for a given firm, in a given year, the negative effect of rising imports costs on exports is more prominent in markets where the firm is smaller relative to other firms in the same sector. In the light of our model, this result implies that the elasticity of markup for a given firm in a given year, is increasing on its market power in the destination market. Intuitively, a multi-destination exporter decides to adjust relatively more its markups (and less their prices and export revenues) in those markets where it has higher market power.