Biggs and Wedges first appeared in Final Fantasy VI, as Terra’s Magitek wingman. They’re lackeys of the Gestahl empire who accompany Terra to make sure she doesn’t regain consciousness from her mind control. They need Terra to find an Esper, from Narshe. When they find Valigarmanda (or Tritoch) frozen in ice, something weird happens. Biggs and Wedge vanish from sight in a flash of light.
From then on, what happens to them is a mystery. Where do they go? Maybe their consciousness travels into an alternate dimension or a parallel universe. Maybe they become the main variations of Biggs and Wedge we see before us today.
We see them in Dissidia as tutors who remember their past. Same with Brave Exvius Final Fantasy. Otherwise, they appear again as soldier/guard types (Final Fantasy IV, VIII, X, XII, XV).
But if they do reincarnate throughout the series, they may have stopped at Final Fantasy VII.