Korg provides a firmware updater tool which does all the magic for you. It requires to turn the Minilogue XD into »Update mode«. Unfortunately, unlike advertised, neither the english nor the german manual provide any hint how to accomplish that.
On youtube I found a video explaining how to do it for the Minilogue – hold the buttons 6 and 8 pressed while powering on the device. Consequently, it’s just buttons 14 and 16 on the Minilogue XD.
After the last update, the Minilogue XD complained during startup
According to a Gearsluts posting, the Panel update can be performed by holding the buttons 14, Rec, and Rest while powering the on device.
I found no information for the Voice update but for the Prologue, which is buttons 5, Exit, and Shift. Translated to the Minilogue XD this turned out to be buttons 13, Rec, and Rest.