Yes - the options for configuring MIDI OUT are fairly well matched to the Korg.

Beyond volume, note on/off, and other "standard" MIDI outputs - what the knobs and buttons do between the two keyboards is different. So there may be editing on the VST side to assign a particular knob to a particular VST parameter - as I'm sure you had to do with the Korg at some point. Lots of VSTs have "learn" features.

I'm not sure if the Korg has different modes for the knobs/buttons/sliders. For knobs and sliders on Montage - there are several "modes" available which can affect the MIDI output of each. Therefore, there is some need to become oriented to this and avoid, during a performance, changing modes unintentionally if that is going to affect your VST control. Keep in mind that changing these "modes" is not easy to do and you are unlikely to, at a gig, foul something up. Some are easier to "trip over" than others - so the orientation step is a good time to learn how these "modes" are switched and relate this to your propensity to inadvertently change the "mode" for a given control.