Seam bowler Carter, 27, joined from neighbours Nottinghamshire last September on a two-year deal.
He took 20 wickets for Derbyshire in 16 appearances across all formats.
Neither Knight, 23, nor Elstone, 26, who are both all-rounders, have made a first-team appearance for Derbyshire this season.
Meanwhile, former Derbyshire and England all-rounder Dominic Cork is to work with the current squad as a bowling consultant as they try to secure quarter-final places in the T20 Blast and One-Day Cup.
"Dominic is an experienced cricketer who knows how to win, claiming silverware with both Derbyshire and Hampshire during his playing career," cricket advisory director Kevin Dean told the club website.
Derbyshire are fifth in the T20 Blast North Group with one game to play, with only the top four progressing to the knockout stage.
They began Sunday's One-Day Cup match against Notts on top of their group in that competition.
Andy Carter, Tom Knight and Scott Elstone have all left Derbyshire after their contracts with the county were cancelled by mutual consent.