It depends on how you tune. If he tunes to the initial attack, then he doesnt have to worry about compensating. I recommend tuning this way if using a low tuning and playing fast material. If there is more suspended chords, then you will want to tune the note after the attack.
with bass, this effect is more pronounced. luckily bass is usually not polyphonic, so It can be tuned easily.usually bass will sound a bit sharp depending on how hard the bassist picks, if you cant get him to lighten up with his pick hand, you may have to tune using melodyne. I would steer away from pitching down the whole track, cuz it will make some notes sound too flat.
I recommend bassists practice with a tuner and a level meter... Im not a very technical bass player, but I am pretty good at playing in tune. i find it is best to press close to the fret, and to try not to bend the string when fretting a note.