The ESPN Fantasy app allows users to customize their alert settings across all different Fantasy Games.
You can even change specific settings for each individual league.

In the ESPN app settings menu, select Alert Preferences.



From the Alert Preferences menu, you can toggle Game Alerts and News Alerts for each of the Fantasy sports offered on ESPN.com (Football, Baseball, Hockey, or Men's and Women's Basketball)



Game Alerts refer to the results of your Fantasy teams and players. News Alerts cover the real-world news and results for the actual professional leagues.

Scrolling further down on this menu will allow you to customize alerts for each individual Fantasy League you manage.



 Just select any one of your leagues from the list to open a new alert menu. You can turn alerts on or off for categories such as Trades, Add/Drops, Home Runs, etc. 

Changes in alert settings to one league won't affect any others. Alerts settings will have to be customized for each league individually.