Not sure about Belgium, but I have just returned from touring France and Ireland from the UK and the only person on the entire journey that was remotely interested in seeing my passport was an rather officious receptionist at the Ibis Hotel in Paris. Never showed any docs or checked in with customs anywhere.The only cross border flight plan I filed was my initial leg from Scotland to Cherbourg which used up the last of my Skydemon credits, so I didn't file on the way back.
In the 20 odd hours of flight time the only interaction I had with officialdom regarding the rules and flight movements was at Oban where upon landing from Ireland I was advised ( in a very friendly way) that wearing a red life preserver is no substitute for a yellow high vis vest when on the ramp.
Prior to my trip I asked on this forum about the ins and outs of european flying, one reply stated it was really easy and no big deal, he was absolutely right.