The teams swapped linebackers, as well as a swap of late-round picks, on March 14. The Texans traded Lawson to the Jets on Sunday, getting a sixth-round choice in return.
The Dolphins cut McKinney on Monday, Ian Rapoport of NFL Media reports.
McKinney is coming off a 2020 season cut to four games due to shoulder surgery. The Texans made the inside linebacker a second-round choice in 2015.
In six seasons in Houston, McKinney started 77 games, totaled three 100-tackle seasons and made the Pro Bowl in 2018.