It sounds like there is no Assignment clause in coaching contracts. That’s the language that permits trades by letting a club assign the player’s contract to any other club in the league without written consent.

Warning: Comparative Sportsology Ahead:
Due to looser affiliations between clubs, this clause also doesn’t exist in European soccer club contracts. This means they use a system very much like the one in the article. Basically, it amounts to this: 1. old and new club agree in principle to a price to release the player from his contract; 2. player and new club agree in principle to salary, bonus, etc.; 3. Everybody executes and files the appropriate paperwork to make it all official. They have protections in place to make sure nobody gets screwed on a legal formality (e.g. you released me at 3:59 pm, but I now refuse to sign my contract with my new team at 4:02 PM).

The situation for NFL coaching changes would be similar since they are employed under a contract but not subject to the CBA or standard player contracts. The difference is what an NFL team would consider useful compensation (e.g. cash, picks, or even players, versus the Euro standard of cash only) and the fact that Roger Goodell’s pimp hand is relatively strong compared to any European soccer official’s.