To complete a pullup, you need to flex the elbow joint and adduct the humerus. The biceps does nothing to adduct the humerus, whereas the lats are the biggest muscle effecting adduction, so they will be involved in your pullup no matter what. Other muscles that can help adduction, and will do so more (sometimes to their own detriment) when the lats get tired, are pec major, subscapularis, teres major, and triceps. Yes, triceps! I used to get major soreness in my triceps ages and ages ago when I first tried to do lots of pullups, and could not understand why that would be...
I am not quite sure why the lats will take over in the initiation phase when the biceps gets tired or is weak, it might have to do something with joint angles - muscles are stronger at certain joint angles than others, and depending on the joint angle, different muscles can become the prime movers.
Ultimately, the details of the biomechanics don't matter so much in this case. Perform the pullups with good form, and you will strengthen the muscles that help you do that - soon to a point where you will have achieved mastery.