It gives a pretty good explanation of how the release trigger works.
Since release triggers are custom installations, just about any configuration is available. You can have release/release, release/pull or pull/release. And I do believe that it can be set up to select which barrel fires first. It depends on the customizing work you want.
The main purpose of release triggers is to overcome flinches. Apparently, letting go with a finger causes a different sensation than pulling with it. It is hoped that this will avoid a flinch that was learned by pulling a trigger and receiving recoil over many thousands of shots.
Some pigeon shooters like a release/pull trigger set-up because they feel that they can fire the second barrel more quickly that way.
Shotgun Report’s Technoid