Generally, trademark rights exist only in countries where the mark is being used. A trademark has a separate existence on a country-by-country basis. In other words, for the same trademark, the rights of the trademark owner depend on the country at issue.
US laws protect against the improper use of a trademark that causes commercial injury - even if the injured party is not a trademark holder. Commercial injury can be in the form of lost sales in the US and/or reputational injury in the US.
Therefore, if a person in the US holds a foreign trademark, but not a US trademark, it is possible that the person can prevent the improper use of a trademark in the US.
Coca-Cola recently tried this tactic.
Coca-Cola held trademark registrations in India for cola products called THUMS UP and LIMCA, but not in the US. Both products were imported and sold in the US.
A competitor, Meenaxi, held US trademark registrations for THUMS UP and LIMCA for cola products.
Coca-Cola tried to cancel those US trademarks based on injury to Coca-Cola in the US. Coca-Cola argued that it was being damaged in the US by Meenaxi "misrepresenting the source of goods."
Coca-Cola was unsuccessful because it did not prove lost sales in the US. It also failed to prove reputational injury - something more than speculation that a dissatisfied Meenaxi consumer might blame Coca-Cola.
ACTION TO TAKE:
Your foreign trademark registration cannot be used in the US to stop infringement in the US. However, you may be able to establish that a US competitor who owns a US trademark registration is harming you.