Assuming the exporter is in country B, and according to B's laws the export is legally fine.
Country A decides what is - in their opinion - legal or illegal, and what they want to prosecute. So country A could decide by creating appropriate laws that the exporter in country B is acting illegally, and that they want to prosecute them. If they have these laws, and want to prosecute, then they could ask country B to extradite the exporter. B would be extremely unlikely to follow this request because the exporter has done nothing wrong in the eyes of country B. The exporter would then be strongly advised not to travel to country A.
But countries will in many cases not be interested with what happens outside their country. Still, the exporter would be wise to check if the export puts them in any danger. If they had applied and received an export license in country B, you would expect that B would have given advice where they shouldn't export to. Like "You received a license that allows you to export to any country in the world except countries X, Y or Z without breaking the law in B. We recommend that you don't export to R, S, T because such an export is illegal according to the laws of these country. "