We have carried out spectroscopic observations in 4 cluster fields using Subaru's FOCAS multi-slit spectrograph and obtained spectra for 103 bright disk field and cluster galaxies at $0.06 \le z \le 1.20$. Seventy-seven of these show emission lines, and 33 provide reasonably-secure determinations of the galaxies' rotation velocity. The rotation velocities, luminosities, colours and emission-line properties of these galaxies are used to study the possible effects of the cluster environment on the star-formation history of the galaxies. Comparing the Tully-Fisher relations of cluster and field galaxies at similar reshifts we find no measurable difference in rest-frame $B$-band luminosity at a given rotation velocity (the formal difference is $0.18\pm0.33 $mag). The colours of the cluster emission line galaxies are only marginally redder in rest-frame $B-V$ (by $0.06\pm0.04 $mag) than the field galaxies in our sample. Taken at face value, these results seem to indicate that bright star-forming cluster spirals are similar to their field counterparts in their star-formation properties. However, we find that the fraction of disk galaxies with absorption-line spectra (i.e., with no current star formation) is larger in clusters than in the field by a factor of $\sim3$--5. This suggests that the cluster environment has the overall effect of switching off star formation in (at least) some spiral galaxies. To interpret these observational results, we carry out simulations of the possible effects of the cluster environment on the star-formation history of disk galaxies and thus their photometric and spectroscopic properties. Finally, we evaluate the evolution of the rest-frame absolute $B$-band magnitude per unit redshift at fixed rotation velocity.