[Dakota]

I'm confused as to the grounds for your protest.

The catcher may be replaced on the bases by a courtesy runner. The courtesy runner must be an eligible sub who has not yet entered the game and who has not been the courtesy runner for the pitcher. The player who enters as the coutesy runner may still be used as a legal sub for any other player. When a courtesy runner is used, the player run for (in your case, the catcher) plays their normal position the next time the team goes on defense (i.e. catcher).

The catcher may also be replaced on the bases by a substitute. The ASA book does not use the term "pinch runner" - but that is what it is - a sub to run the bases. In this case, the player who is being run for is removed from the game. That player has one re-entry remaining.

The scenario you described (catcher reaches base and is replaced on the bases by another player and the catcher returns to catch the next half inning) is legal under both rules.

Using the courtesy runner rule, the catcher has not officially left the game and has not used a reentry and the courtesy runner may be used as a sub for other players.

Using the substitution rule ("pinch runner"), the runner has been entered in the game as a sub for the catcher and has then left the game, and the catcher has left the game and re-entered to play defense.