Decision feedback equalizer
Let's assume y to be a sequence of individually detected bits and the system in your image tries to catch a multibit pattern (or a class of patterns) from the stream regardless some errors in bit by bit detections. The information about the existence of the interesting pattern has been spred over the whole response of your FFF so you should not discard anything assuming your system is properly designed and does not have fully useless redundant delay.

You will not get the indication of the existence of the interesting pattern as soon as the last bit has been inputted to FFF, but some amount of time later.

Think about a radar which sends continuous signal. Echos from separate targets overlap. Nothing really should be discarded. If you know there's no overlaps, you can obviously discard everything between a detection and the first possible arrival of the first bit of the next interesting pattern.