Linux has long had a problem with programs that hog all the system's "dirty" cache memory. What is happening is that the copy process is filling the write cache with the file data it is copying and it is doing it very quickly. So when Firefox comes along and needs to write it must first wait for dirty buffer space or an available disk queue write slot. While waiting it is competing with the copy process and the kernel's pdflush thread, which moves data from dirty buffers to the disk write queue.
Firefox has yet another problem in this scenario. It uses SQLite to store its bookmarks, history and other things. SQLite is a ACID compliant database and it uses a transaction system with its disk writes flushed to disk. So not only does it have to wait for buffer space, it must wait for the disk queue, which is full of copied file, to clear out before it can acknowledge a successful write.
There has been a lot of tweaking done to the Linux disk queuing and buffering system. There are changes in almost every kernel release. Try one of the newer releases. You can also try tweaking the sysctl values. I sort of like these:
vm.dirty_writeback_centisecs = 100
vm.dirty_expire_centisecs = 9000
vm.dirty_background_ratio = 4
vm.dirty_ratio = 80
You can also try tweaking the number of slots in the disk queue. This value is in /sys/block/sda/queue/nr_requests. You need to substitute sda with whatever your drive really is. More slots means more chances to merge IO requests and the CFQ IO scheduler can do a better job with priorities. Fewer slots usually means a shorter wait to get written to disk for synchronous IO like SQLite's transactions. Fewer slots also means a shorter wait to get read IO into the disk queue if a write-heavy process completely stuffs the queue with write IO.