The presence of distant protoplanets may explain the observed gaps in the dust emission of protoplanetary disks. Here, we derive a novel analytical model to describe the temporal decay of the pebble flux through a protoplanetary disk as the result of radial drift. This has allowed us to investigate the growth and migration of distant protoplanets throughout the lifespan of the disk. We find that Moon-mass protoplanets that formed early on can grow to their pebble isolation mass, between approximately 20 and 80 M_⊕, within less than 1 Myr, in the 20-80 AU region around solar-like stars. The subsequent fast migration in the early stages of gas accretion, after pebble accretion ends, transports these giant planets into their final orbits at <10 AU. However, our pebble decay model allows us to include a new pathway that may trigger the transition from pebble accretion to gas accretion after the pebble flux has decayed substantially. With this pebble decay pathway, we show that it is also possible to form gas giants beyond 10 AU. The occurrence of these wide-orbit gas giants should be relatively low, since their core must attain sufficient mass to accrete gas before the pebble flux decays, while avoiding excessive migration. Since these gas giants do not reach the pebble isolation mass, their heavy element content is typically less than 10M_⊕. Our results imply that the observed gaps in protoplanetary disks could be caused by distant protoplanets that reached the pebble isolation mass and then migrated, while gas giants in wide orbits, such as PDS 70 b and c, accreted their gas after the decay in the pebble flux.