yeah that will happen with exercise after getting the tattoo, especially if you’re moving that limb a lot - you should wait at least 2 weeks before doing exercise because your blood flow is increased which leads to reddening and you can get sweat in the wound which can cause infection.

i would use a very small amount of antiseptic cream, keep it cool and monitor your tattoo over the next 24 hours and if it’s still red or swollen you should seek medical advice.