Almost everything the mobile apps pull from the Canvas API is in sync with Canvas web. There are a couple of exceptions where server caching causes a delay, like with discussion replies. If a person posts a discussion reply in Canvas web, that may only show up in the API a couple of minutes later when the cache refreshes. But almost everything is instantaneous (e.g., you change a due date in Canvas web, the next second a student loads that assignment in the app and sees the updated due date). I haven't heard about any caching issues affecting quizzes.
As far as push notifications go - they're sent as fast as possible, though if you happen to change an assignment at the same moment that a thousand other instructors change an assignment, the push notifications queue up and may take a few minutes to process. We have a combination of automated processes and humans to handle queue spikes, so the time between an update to Canvas web and the resulting push notifications should either be short or instantaneous.
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