Specifically, by what time was the stereotypical imagery and professionalism associated with the Legions, such as tight formations with large square shields, the gladius and pilum, and red color scheme no longer applicable? Or were changes too fluid to pin down such a time? I know, at least, that such images were never associated with the Byzantine military despite them being a "continuation" of Rome.
After or during the crisis of the third century. Basically the rough outline was that the crisis caused roman legions to be split apart and rushed all over the Empire dealing with emergencies. As a result this was why legion sizes dropped from 6000 ish men to around 500-1200. Naturally this is disputed. Late roman unit sizes are incredibly contested. Now. In terms of say things like equipment. Things did change. Segmentata seems to have been phased out. The scutum and gladius as well. The hasta and oval or round shields made a larger appearance. Also. Calvary made up a vastly more notable role in the late army.
Sources.
De re militari by vegetius
"A History of the Byzantine state" by Warren Treadgold
"Byzantium and it's army 284-1081" by Warren Treadgold
"The Byzantine State" by Ostrogorsky
"History of the Byzantine Empire" by Vasiliev