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Does "speed up at the beginning of their expansion" mean something different than "increase in velocity from beginning" to you? I don't know what you're asking that I haven't already answered before you asked it. "Speed up" and "velocity increase" seem synonymous in this case.
So say I was holding a rod made of the universes lightest material, the rod is 99999999999999999999999999999999999999999999999999999 KM long, and when you pick up one end of a stick the other end seemingly also reacts instantaneously thus is the nature of connection.