with block inlays and or white binding, do you tape it of? It seems that, that would become rather tedious on guitars with cloud or vine of life inlays, or any odd shaped inlays.
Usually, it is stained before the inlays are actually installed. But if they are already installed, you meticulously clean the stain off of them afterwards.
I always wonder this too when people talk about darkening a fretboard with anything even shoe polish. I saw somewhere this being done and there were telltale signs like scratches that weren't visible in the white areas which got permanently stained dark brown afterward...looked bad. But the thing is, it all depends on the stain or finish and what the inlays are made of. You know how people always want to age white plastic parts that look too new? Some its as easy as soaking in coffee for a few hours. But I have one set of Gotoh white plastic tuner buttons that could soak in used motor oil and still look like Antarctica afterward. I didn't go as far as dousing them with actual dye though.
Also, most stains will wipe off of light colored inlays easily as long as you don't let the stain dry on them. You should be applying the stain and wiping it soon after anyway.
The nice thing about that stain pen is you can avoid saturating the entire inlays since you can kind of "draw" around them. Regardless, it's rare that modern inlay materials are adversely affected by brief exposure to stains like this, though it's always possible you'll get the slightest hint of darkening (which is nothing compared to how much darker the board gets, so they end up looking normal anyway).