Global events can be activated from any security zone in the Season menu.
If you want to participate in a global event, you must first be in a safe area. In fact, scratch that: you need to check if it is really active. Fortunately, you can do both of these steps simultaneously. In a safe area (Safe House or Base of Operations - friendly checkpoints do not count), open the Season menu from the main menu. You should notice the roadmap in the center: scroll down to the global event (it will literally say “global event”) and see if it's active. If it's great, you can participate.
From there you need to select Open, or if you've pulled up the description, select Details (these should be mapped to the same button). Once inside, you'll see a toggle at the bottom of the screen. If you are not in an appropriate safe zone, it will be grayed out. All you need to do is toggle it to active and voila, the global event will be activated.
There are a few things you should consider when activating a global event. First of all, this will affect matchmaking, so if you've enabled it, you'll only match with groups that have it enabled as well. Second, you can activate it for yourself if you are in a predefined group. I know these two points are contradictory, but after a lot of testing it works. That said, your teammates will see the unique Global Event icons above enemies' heads, but they won't be affected by the Global Event modifier.
If you want to track your challenges and rewards, these can be found in the same menu as the activation toggle, and you need to manually redeem your rewards. Whenever you want to deactivate a global event, simply repeat the process used to activate it.
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