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Not 100% clear what router buffers you referring to that became full.
About the closest feature that a router might use to notify the source to slow its transmission rate is ICMP source quench, but that a) not tied to router performance, as far as I know (although it could be tied to output queue drops) and b) generally not used anymore for various reasons.
As to why the sender slowed, if the sender was using TCP, it would slow its rate if it also saw packet drops and/or had to wait on ACKs.