The remarkable progress in the field of laser spectroscopy induced by the invention of the frequency-comb laser has enabled many high-precision tests of fundamental theory and searches for new physics. Extending frequency-comb-based spectroscopy techniques to the vacuum and extreme ultraviolet spectral range would enable frequency measurements of transitions in, e.g., heavier hydrogen-like systems and open up new possibilities for tests of quantum electrodynamics and measurements of fundamental constants. The two main approaches, full-repetition rate up-conversion in a resonator and two-pulse amplification and up-conversion for the Ramsey-comb technique, rely on high-order harmonic generation (HHG), which is known to induce spurious phase shifts from plasma formation. After our initial report [Phys. Rev. Lett. 123, 143001 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.143001], in this article we give a detailed account of how the Ramsey-comb spectroscopy technique is used to probe the dynamics of this plasma with high precision and enables accurate spectroscopy in the vacuum ultraviolet. It is based on recording Ramsey fringes that track the phase evolution of a superposition state in xenon atoms excited by two up-converted frequency-comb laser pulses. In this manner, phase shifts up to 1 rad induced by the HHG process could be observed at nanosecond timescales with mrad-level accuracy at 110 nm. We also show that such phase shifts can be reduced to a negligible level of a few mrad. As a result, we were able to measure the 5p6→5p58s2[3/2]1 transition in Xe132 at 110 nm (the seventh harmonic of 770 nm) with sub-MHz accuracy, leading to a transition frequency of 2726086012471(630)kHz. This value is 104 times more precise than the previous determination and the fractional accuracy of 2.3×10-10 is 3.6 times better than the previous best spectroscopic measurement using high-order harmonic generation. Additionally, the isotope shifts between Xe132 and two other isotopes (Xe134 and Xe136) were determined with an accuracy of 420 kHz. The method can be readily extended to achieve kHz-level accuracy by increasing the pulse delay, e.g., to measure the 1S-2S transition in He+. Therefore, the Ramsey-comb method shows great promise for high-precision spectroscopy of targets requiring vacuum ultraviolet and extreme ultraviolet wavelengths.