While the standard scenario of third-harmonic generation (THG) by a dispersive-wave pump involves the emission of light with a frequency 3omega, thrice the frequency omega of the input pump field, solitons undergoing a continuous shift of their central frequency omega due to the Raman effect in a multimode optical fiber can generate the third harmonic in a different fashion. In the experiments reported here, we provide the first direct experimental evidence of THG by a continuously red-shifting soliton pump by studying the third-harmonic buildup in relation to the spectral evolution of the soliton pump field in a silica photonic-crystal fiber (PCF). We show that solitons excited in a PCF by unamplified femtosecond pulses of a Cr:forsterite laser sweep through the spectral range from 1.25 to 1.63 microm , scanning through a manifold of THG phase-matching resonances with 3omega dispersive waves in PCF modes. As a result, intense third-harmonic peaks build up in the range of wavelengths from 370 to 550 nm at the output of the fiber, making PCF a convenient fiber-format multifrequency source of short-wavelength radiation. Time-resolved fluorescence measurements with photoexcitation provided by the third-harmonic PCF output are presented, demonstrating the high potential of PCF sources for an ultrafast photoexcitation of fluorescent molecular systems in physics, chemistry, and biology.