Does a players weight fluctuate at all based on the player's position and experience at the position?

Yes, players' weight will trend toward the normal weight for their assigned position.

Thank you! I thought so based on experimenting with the game, but I had not seen it documented anywhere. I assume there is a cap for this. For instance, I don't think a FB would ever get as heavy as a RG, etc., correct?

Each position has an ideal weight, so based on the assigned position they will trend toward that ideal weight.

I've got a player who was 12% underweight and noticeably faster. Will he slow down as his weight trends up? Can I get him to stay light?

Depends on what position he's playing.
EG; You have a RB listed as 225lbs but you change his position to WR, slowly his weight will decrease to an 'ideal' WR weight, conversely that works the other way too.
Say that RBs speed is 80, it won't go up if you make him into a WR, but because he is now lighter he will seem faster

It's my punt returner RB G. Begay. He has 100 speed and 100 accel. I'm worried that as his weight creeps up his super speed (and the reason I drafted him) will become just normal RB 100's. He came in at 187lbs I believe and is already up to 190. Pretty soon he'll eat his way out of a job.

As his coach/GM I want to be able to say keep your weight the same and if its a durability issue I'll limit carries rather than lose speed.

In that case (if I were you) I would convert him to a WR, you can still play him at RB if you wanted to as well, but his primary would be a WR build. Not sure about this but its possible other attributes may be affected like strength, avoid fumble and ball carrying. IMO KR & PR don't need all those attributes, just a good Kick Catch

I agree with Davesgang.

I don't think it makes any sense for all players at the same position to eventually be the same weight.

Are they????