By exploiting the excellent short-term phase stability between consecutive pulses from a free-running optical parametric oscillator frequency comb, we report the first example of hollow-core fiber-delivered heterodyne spectroscopy in the 3.1-3.8 µm wavelength range. The technique provides a means of spectroscopically interrogating a sample situated at the distal end of a fiber, with all electronics and light sources situated at the proximal end and with an inherent capability to suppress spectroscopically interfering features present in the free-space and in-fiber delivery path. Using a silica anti-resonant, hollow-core delivery fiber, we demonstrate high quality transmission and attenuated total reflectance spectroscopy of a plastic sample for fiber lengths of up to 40 m, significantly exceeding the few-meter lengths typically possible using solid-core fibers. The technique opens a route to implementing multi-species spectroscopic monitoring in remote and / or hostile industrial environments and medical applications.