Indeed you don't NEED an DSLR to take birds in flight, though they are definitely faster and better designed for such shots...and not all P&S cameras are equal, so depending on your Fuji model, you may or may not have a good tool for such shots.
However, as also mentioned, even folks with DSLRs can struggle with birds in flight - it does require some practice, some skill, and some experience to really start to get right...quite a few people shoot for years until their skill has risen to the level where they can reliably shoot birds in flight. A DSLR can help make it easier, or increase the hit rate, IF you understand what you're doing, have the right settings, are using a good enough lens, and have good tracking/panning technique and timing. This is because the phase-detect focus systems are faster than typical contrast-detect focus systems used by fixed-lens compacts, and because the phase-detect AF systems can better track motion that is closing or parting from the camera's position. With practice and technique, good timing, and decently sized birds, a P&S can take bird in flight shots - but isn't likely going to work well with tiny sparrows at full speed...even DSLRs can struggle there. Seagulls, pelicans, large waders, etc you can shoot with a P&S with practice and timing - but don't expect a 100% hit rate. Even DSLR birders sometimes fire off bursts of shots of which some missed focus a bit or they didn't keep things in frame well enough - so maybe just one or two out of a sequence are keepers.
