Really nice!
Refresh my memory - these aren't battle cavalry right? Their primary function is like that of Military Police? Going after deserters and all that?
Their primary function was to guard the imperial palaces and mansions, and also to provide an escort for Bonaparte as well as French and foreign dignitaries. But they did fight as battlefield cavalry every now and then : in Spain at Medina de Rio Seco, during the Leipzig campaign and also in 1814, notably at Montmirail. They were also at Ligny and Waterloo during the 1815 campaign.
Chasing deserters and maintaining order was something they occasionally did, as the print below shows :

Because of their guard status, however, they often left these duties to the regular
Gendarmerie. The elite
gendarmes also provided security for the imperial headquarters and were tasked with protecting trophies of war.