Border carbon adjustments imply that high-income countries set taxes on energy-intensive imports that are proportional to the carbon content of these imports, to match their own carbon taxes. This paper considers the impacts of such a policy on exporter countries, many of which have no or very low carbon taxes today. The paper first studies a policy whereby the importer allows the exporter's border tax to be reduced by its own comprehensive carbon tax ("tax rebating"). The analysis finds that the exporter is then incentivized to set its own comprehensive carbon tax at the same rate as the border tax, up to a maximal rate. When the border tax is higher, the exporter instead reduces its carbon tax. Border tax revenues of the high-income country can be returned to incentivize higher carbon taxes in the exporting countries ("carbon crediting"). When tax rebating is not allowed but tax revenues are fully returned, even higher exporter carbon taxes can then be incentivized, possibly exceeding 0 per ton of carbon dioxide in the numerical examples. Border taxation can give rise to export diversion away from border tax-setting countries, which reduces the scope for incentivizing the exporter's carbon tax. The paper also studies how taxes on oil extraction by oil exporters can be incentivized by oil importing countries, by increasing their oil import prices above world market rates, or more efficiently through support to investments in exporters' renewable energy capacity.