The first year's membership was tightly constructed.
Dr. Light wasn't intended to be a full member, at least not yet. Her story purpose was to get the League into the UN.
Booster Gold's joining (editorially mandated, as were the founders) was used to serve the Maxwell Lord plot, as did Dr. Light's initial membership.
The League, once Giffen and DeMatteis got started, was slated to turn international before the end of the year, as part of the Maxwell Lord storyline, with two new members added to reflect it.
Dr. Fate was removed as one of the slots to free up for Captain Atom and Rocket Red, as Giffen and DeMatteis already had plans for a solo title.
Captain Marvel was removed to make room for Captain Atom and Rocket Red, likely because he was just the least interesting character dynamic at the time and was probably 'voted out' by Giffen and DeMatteis. Although, IIRC, there were plans for a Captain Marvel title that never materialized.
Rocket Red joining also sync'd well with Millennium, as they would need a Manhunter in the book.
And the replacement Rocket Red connected with the early issues, which strengthened the book's internal continuity.
Blue Beetle was almost taken away during Millennium, as Wein originally had plans to relaunch his title with a new Blue Beetle. But when plans changed, Beetle was spared and remained in the JLI.
After the first year, though, Giffen and DeMatteis were a bit less in control of the comings and goings.
Black Canary was removed due to Grell's use in Green Arrow, which also left the team with no females... So Fire and Ice were brought in.
Wonder Woman was taken away before the first issue hit the stands, requiring the previously solicited cover to #3 to be edited with her replaced with Catherine Cobert.
Batman rarely appeared after the second year, and was likely being kept out due to editorial's preparation for the movie.
Animal-Man left after the events of Morrison's Animal-Man #20.
Huntress was given to them by Andy Helfer, who edited the JLI titles and the recently cancelled solo Huntress book. It would seem that Giffen and DeMatteis just couldn't find a purpose for her on the team and they struggled to find a good voice for her that interacted well with the others.
JLI #42 made light of how characters were blocked to Giffen and DeMatteis, with several then-popular characters being offered membership and flatly turning it down.
And most, if not all, the new members post-Huntress were characters that were free of other writers or editors: Bluejay, Silver Sorceress, Crimson Fox, Lightray, Orion and General Glory.