There are many things that could be going wrong, but in my experience, not swinging (or having a bad swing feel) is usually a result of small technique-related issues. In particular, it can include: (a) not playing the different pieces in unison, (b) playing the triplet rhythms unevenly, and (c) playing certain pieces too loud. All of these things can be cured with practice (starting at slow tempos and gradually increasing).
To try and diagnose the issue, you can have the drummer play at 80 bpm with the ride cymbal only on quarter notes and the high hat on beats 2 and 4. If the left hand and right hand are striking in time together, it should be easy to imagine the eighth notes--and it should sound swinging. It should also sound controlled: the volume should be even and there shouldn't be random notes that are played more loudly due to weak technique. Next, add in the feathered bass drum. The drummer probably doesn't play four to the floor, but go ahead and do this to ensure the bass drum isn't interrupting his hands' timing. You also want to listen for the bass drum being too loud. Not feathering the bass lightly enough (e.g., because the drummer comes from a rock setting, etc.) can hurt the feel.
Once you and the drummer have identified that this sounds good, add in eighth notes on the ride. If the drummer doesn't have a rigid practice routine, the triplets could very likely be uneven. Putting the eighth note triplets at the wrong spot rhythmically can really hurt the feel. Check to see if the triplet rhythms are being played evenly/correctly by turning the metronome on and adding clicks for the eighth note triplets. The ride pattern should be exactly in unison with the metronome. Turn up the metronome 5-10 clicks and make sure the ride is still right with the eighth note triplet beat. Continue increasing the metronome like this until you've reached the tempos of the fastest tunes you play.
You can have the drummer add in the snare, but he's probably playing something more modern-sounding than just 'snare on beat 4.' The genres you're playing are different from older swing/big band music, and so part of the issue could be stylistic. In my opinion, the single best thing a drummer can practice for a bebop/post-bop feel are Elvin Jones inner triplets. To quote a small passage from Jon McCaslin's site, which goes more in depth:
The first patterns that Elvin demonstrated deal with different ways that he voiced inner triplet subdivisions around the drum set in the context of the jazz ride cymbal rhythm. By exposing and giving more attention to those inner triplets, it serves to help open up and stretch the beat quite a bit. This is something Elvin was renowned for.
Play the following pattern with the Right hand on the ride cymbal, filling in the missing triplets on the snare drum with the Left hand:
The basic sticking pattern resembles this:
RLL RLR RLL RLR RLL RLR RLL RLR etc.
He goes more in depth and has additional exercises. This type of inner triplet practice (starting at very slow tempos and gradually increasing) creates a great feel for the styles you're playing.
Here's a video that goes through the process of stripping down the feel and adding elements back in one piece at a time:
Here's a phenomenal example of the Elvin Jones inner triplets at a medium tempo (from 0:42 to 1:45), to give a sense of what the inner triplet exercises are working toward and why they're valuable to practice thoroughly: