MIUI likes to kill apps that run in the background to save battery life but many users don't like this strategy because this way they miss important notifications. Thankfully it's possible to disable most of these countermeasures.
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Go to Settings > Battery & performance > Choose apps and set "No restrictions" for all your important apps.
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Go to Settings > Battery & performance > Scenes and disable all strategies here.
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Go to Settings > Permissions > Autostart and enable autostart for the apps you'd like to receive notifications from.
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Go to Settings > Notifications & status bar > App notifications and tweak the notification settings for each app.