Why did she say "Are you bringing ~" not "Do you bring ~"?
To answer this question, it is important to understand the tense difference between these two phrases. "Are you bringing" is in the present continuous tense and "Do you bring" is in the present simple tense. The present continuous tense refers to things that are currently happening. The simple present tense is used to discuss things that are done all of the time. Ellen uses the present continuous tense because she is asking if the cast currently brings their children to the Fuller House set. If she said "Do you bring your children on set" Ellen would be asking if they habitually bring their children to the sets they work on, not just the Fuller House set.
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