She flew down the stairs, her children tumbling after her.
This absolute tells us something about the circumstances attending the lady's rush downstairs, but it doesn't modify anything in the main clause, nor is it an object or a subject. It simply is not a grammatical part of that clause. (The term absolute derives from a Latin word meaning "free, unrestricted.")
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What you need to know about… Project Management Made Easy! Project management consists of more than just a large building project and can encompass small projects as well. No matter what the size of your project, you need to have some sort of project management. How you manage your project has everything to do with its outcome.