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or does Dropbox actively push any changes to all devices?I may be misunderstanding your question, but it seems to me that this is literally the whole point of the service. Dropbox is a folder that syncs. You change something on one device (one write), it updates everywhere (multiple reads).
They do have some tricks to lower their bandwidth usage. AFAIR, if multiple devices are in the same LAN, only one gets synced with the cloud, and the rest sync to that one. Also, they have (recently introduced) a sync option, in which fake files are visible in your file system, but no actual contents get put on the drive until you try to open them.