It would make sense for a hash to be sent as a first step. The cloud will always have the most up-to-date information on objects which can be classified as:
Known safe: leave the object or scan once and leave (varies from vendor to vendor).
Known malicious: destroy the object, inform EFR which will take the object timestamp, check for additional objects created around this time (+- some offset of 20-30 seconds), and destroy all related objects. These additional objects can (and most likely will) include browser caches. Provided object was not executed, that will be all.
Unknown: object will be scanned using the full set of Kaspersky capabilities, including static analysis, dynamic analysis and definitions. The object will then be classified as "safe" or "malicious". If malicious, EFR will be informed.