Just my .02, could be a couple of things. One of them could be your heavy presses catching up with you and causing you some shoulder pain later on in your session. Another could be simply using too much weight for flies, as I think most people do. Are you using cables or dumbbells? Personally, I’ve found flies work better for me (better pump, no shoulder pain, easier to initiate movement with pecs and keep the tension there) with cables and pec deck.
Be sure whichever version you’re doing, you’re using a weight that allows you to initiate tension and movement from the pecs and get a great squeeze. You should feel like you’re flexing your pecs, and your arms are just an extension of your pecs. If you feel like you’re moving the weight from your shoulders/arms, especially biceps, try going lighter. I think flies work best in a higher rep range, (anywhere from 10-15) as lower rep ranges tend to use more weight, and it’s just harder to keep flies in the pecs with that much weight. Again just my thoughts!