Aims. Migration of dense gaseous clumps that form in young protostellar disks via gravitational fragmentation is investigated to determine the likelihood of giant planet formation. Methods. High-resolution numerical hydrodynamics simulations in the thin-disk limit are employed to compute the formation and long-term evolution of a gravitationally unstable protostellar disk around a solar-mass star. Results. We show that gaseous clumps that form in the outer regions of the disk (>100 au) through disk fragmentation are often perturbed by other clumps or disk structures, such as spiral arms, and migrate toward the central star on timescales from a few thousand to few tens of thousands of years. The migration timescale is slowest when stellar motion in response to the disk gravity is considered. When approaching the star, the clumps first gain mass (up to several tens of M_Jup), but then quickly lose most of their diffuse envelopes through tidal torques. Part of the clump envelope can be accreted onto the central star causing an FU-Orionis-type accretion and luminosity outburst. The tidal mass loss helps the clumps to significantly slow down or even halt their inward migration at a distance of a few tens of au from the protostar. The resulting clumps are heavily truncated both in mass and size compared to their wider orbit counterparts, keeping only a dense and hot nucleus. During the inward migration, the temperature in the clump interiors may exceed the molecular hydrogen dissociation limit (2000 K) and the central region of the clump can collapse into a gas giant protoplanet. Moreover, migrating clumps may experience close encounters with other clumps, resulting in the ejection of the least massive (planetary-mass) clumps from the disk. We argue that FU-Orionis-type luminosity outbursts may be the end product of disk fragmentation and clump inward migration, preceding the formation of giant protoplanets on tens of au orbits in systems such as HR 8799.