Resonant two-photon ionization spectra of the 1-DELTA(u) <-- 1-SIGMA(g)+ transition of C4H2, C4HD, and C4D2 cooled in a supersonic molecular beam have been recorded and partially analyzed. The cooling provided by the supersonic expansion reveals several vibronic transitions not resolved at room temperature. While some progress is made on this basis in the vibronic assignments, the full analysis of the spectra is hindered by the considerable complexity of the upper state spectroscopy involving both Renner-Teller and Herzberg-Teller coupling. Only vibronically induced transitions are observed in the spectra. The absence of the 2(0)n transitions indicates that the vibronic spectroscopy can be treated assuming a linear or quasi-linear upper state geometry. The observation of the 2(0)n transitions in past matrix studies must be ascribed to matrix effects. The narrowing of the rotational band contours in the molecular beam allows a study of the single vibronic level broadening inherent in the transitions. The full width at half-maxima of the transitions in C4H2 increase from 5 cm-1 at 6(0)1 to 40 cm-1 at 2(0)(2)6(0)1, indicating strong, energy-dependent coupling of the vibrational levels in the 1-DELTA(u) state to a dense bath of background levels. The widths of corresponding transitions in C4H2, C4HD, and C4D2 generally decrease with increasing deuteration, indicating that H-atom motion plays an important role in the coupling. There are also dramatic differences in width from one vibronic level to the next. Some suggestions are made as to the source of these differences.