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Low mass X-ray binaries (LMXBs) show strong variability over a broad range of time scales. The analysis of this variability, in particular of the quasi-periodic oscillations (QPO), is key to understanding the properties of the innermost regions of the accretion flow in these systems. We present a time-dependent Comptonisation model that fits the energy-dependent rms-amplitude and phase-lag spectra of low-frequency QPOs in black-hole (BH) LMXBs. We model the accretion disc as a multi-temperature blackbody source emitting soft photons which are then Compton up-scattered in a spherical corona, including feedback of Comptonised photons that return to the disc. We compare our results with those obtained with a model in which the seed-photons source is a spherical blackbody: at low energies the time-averaged, rms and phase-lag spectra are smoother for the disk-blackbody than for a blackbody, while at high energies both models give similar spectra. In general, we find that the rms increases with energy, the slope of the phase-lag spectrum depends strongly on the feedback, while the minimum-lag energy is correlated with the disc temperature. We fit the model to a 4.45-Hz type-B QPO in the BH LMXB MAXI J1438-630 and find statistically-better fits and more compatible parameters with the steady-state spectrum than those obtained with a blackbody seed-photons source. Furthermore, we successfully apply the model to the type-C QPO in the BH LMXB GRS 1915+105, and thus conclude that this variable-Comptonisation model reproduces the rms and phase-lags of both type B and C low-frequency QPOs in BH LMXBs.