I'm assuming you have other sites connected to your MPLS network beyond sites A and B. I further assume you don't have anything like PVCs across your MPLS network; any site can communicate with any site.
The best way to deal with this situation, not only for high to low, but for multiple sites to one site, is QoS on the MPLS egress link. Normally, you won't have much control beyond marking traffic to work within the provider's QoS model. Mark FTP into a bulk class, and only that class should see a major increase in latency.
PS:
The other direction, low to high, should not only mark to take advantage of the MPLS provider's QoS model, but should implement congestion management for its outbound traffic. FTP again would be managed to not be detrimental to other traffic.
Summary: manage local congestion, mark traffic to manage remote egress congestion.