Different sport, different muscle stimulation and recruitment, different efficiency, same general energy systems more or less.
This is why someone who can run marathons who has never boxed in his life can't last 1 round of boxing, and vice versa. Their bodies aren't adapted to the training and need time and constant practice to become efficient. This also applies to the grappling/striking argument.
Your body will always strive to be efficient, this is why you throw different things at it. When you condition, you don't want your body to be efficient in a movement, you want the energy system you are using to be efficient by pushing yourself. If you are already extremely good at a non sport-related movement, it is time to move on.
So. Yes, outside conditioning does affect your performance in a particular sport, but to a certain extent. This is why your conditioning should always be changed up so your body wont be able to adapt, you are not trying to be a master of calisthenics or a powerlifter or an olympic sprinter, you are trying to be a master of your sport first and foremost. So the bottom line with conditioning would be, pushing yourself.
With general conditioning, you are training different energy systems, which gives you overall conditioning as opposed to sport specific conditioning.
Yes it is worth doing, if you are a fighter you need every advantage you can get over your opponent. With running and other constant movement aerobics you can make your recovery between rounds faster, thus letting you go for a whole 12 rounds.
With anaerobic and explosive movements, you will be able to sustain multiple explosions for a set amount of time and not gas afterwards.
Without proper conditioning, you wont be able to access your fighting skills and impose them on your opponent. All your skills will be locked out by something called fatigue. You need to be able to use your skills to their fullest throughout the whole contest, and this is why we condition.
I'm probably preaching to the choir though, but focus on your sport most importantly and keep conditioning a big part of it, just don't get hung up on being a master of a certain conditioning movement, become a master of getting yourself in shape for your sport.