You should always play with darts from the league you are currently in, as you progress up the leagues darts from previous leagues become less stable, meaning less accuracy.
You can see how your current darts will perform in your current league at a glance by seeing what kind of emoji is displayed next to the stability stat of your darts.
A smiling face is Good, you don't need to upgrade. A neutral face is okay, you should consider upgrading. A sad face means you should definitely upgrade. In general you should try and stay above 85-90.
In battle mode darts will perform as if in the correct league.