In the official editor, there's multiple tabs visible after loading the database.
Go to the "competitions" tab, then select the league you want the teams to be in.
In the "teams" section, you'll see the teams currently playing in that league.
When you highlight a team (left click the arrow to the left of its name), the "swap" button at the bottom lights up.
Press the "swap" button, and you can select a team to replace the highlighted team in that league. The highlighted team will move to the latter one's original competition.
Repeat this for all the teams you want in the league, and any other league placed below (or there will be unwanted teams syphoning in after each season).
The online bit is easy, only the host needs to have the altered database.
I'm not really sure what programming screen you say we're talking endlessly about?