We report results of a systematic study of one-dimensional four-wave moving solitons in a recently proposed model of the Bragg cross-grating in planar optical waveguides with the Kerr nonlinearity; the same model applies to a fiber Bragg grating (BG) carrying two polarizations of light. We concentrate on the case when the systems spectrum contains no true bandgap, but only semi-gaps (which are gaps only with respect to one branch of the dispersion relation), that nevertheless support soliton families. Solely zero-velocity solitons were previously studied in this system, while current experiments cannot generate solitons with the velocity smaller than half the maximum group velocity. We find the semi-gaps for the moving solitons in an analytical form, and demonstrated that they are completely filled with (numerically found) solitons. Stability of the moving solitons is identified in direct simulations. The stability region strongly depends on the frustration parameter, which controls the difference of the present system from the usual model for the single BG. A completely new situation is possible, when the velocity interval for stable solitons is limited not only from above, but also from below. Collisions between stable solitons may be both elastic and strongly inelastic. Close to their instability border, the solitons collide elastically only if their velocities c1 and c2 are small; however, collisions between more robust solitons are elastic in a strip around c1=-c2.
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