Playback for different parts coming in late or early

I'm writing a military march for concert band (transferred from Muse 3 to 4 if that important to know). I've only had one issue while writing it so far: whenever I play back the same part on different instruments (the same music on different parts, I guess you could say), the instruments don't always line up with each other or with the tempo of the piece. For instance, the melody I establish after the key change is played by trumpets, piccolo, flutes, and clarinets. The trumpets are very on beat. The flutes are a tad late at some parts of the melody. And the clarinets are even more noticeably late. I did copy-paste the parts from one to the next, but this still shouldn't be happening. It makes a difference because in busier sections of the song it can sound really muddy if the parts playing the same part aren't staying together and on the beat.