A good solution:
Make movement speed be determined by how many items are in the container. A full container would make you move the slowest. The criminal would be required to take more BB drain while carrying the container off of the claim, or be forced to remove some items from the container before being able to move it at a reasonable speed, which would proc the braziers and leave more scents.
As for complaints about carrying containers for non criminal purposes...well they arent really meant to be carried very far anyways, that is what a sled is for. If you want to move your container quickly, empty the contents of it first.