  We report results of systematic investigation of dynamics featured by moving
two-dimensional (2D) solitons generated by the fractional nonlinear
Schroedinger equation (FNLSE) with the cubic-quintic nonlinearity. The motion
of solitons is a nontrivial problem, as the fractional diffraction breaks the
Galilean invariance of the underlying equation. The addition of the defocusing
quintic term to the focusing cubic one is necessary to stabilize the solitons
against the collapse. The setting presented here can be implemented in
nonlinear optical waveguides emulating the fractional diffraction. Systematic
consideration identifies parameters of moving fundamental and vortex solitons
(with vorticities 0 and 1 or 2, respectively) and maximum velocities up to
which stable solitons persist, for characteristic values of the Levy index
which determines the fractionality of the underlying model. Outcomes of
collisions between 2D solitons moving in opposite directions are identified
too. These are merger of the solitons, quasi-elastic or destructive collisions,
and breakup of the two colliding solitons into a quartet of secondary ones.
