For sure - right that way. Belonging to what they actually are (waiting, travelling or whatever) with more baggage and even baggage organized differently on the saddle or beside their (lost?) horses doing this or that.
And i forgot to mention chinese & medieval russian and european cavalry like norman, carolingian and later one knights.
Maybe even india, samurai, scythian and gothic cavalry, (greeks including macedonian, persian including all the different dynasties as rome includes all troop-types from italic to sarmatian from republic to byzanthine).