The low-energy lattice dynamics of the quasiperiodic spin-ladder cuprate Sr14-xCaxCu24O41 are investigated using terahertz frequency synchrotron radiation. A high density of low-lying optical excitations is present in the 1-3 THz energy range, while at least two highly absorbing excitations stemming from long-wavelength acoustic oscillations of the incommensurate chain and ladder sublattices, are observed at subterahertz frequencies. The effects of Ca substitution on the subterahertz quasiacoustic sliding mode gaps is investigated using coherent synchrotron radiation. Analysis of the results suggest increasing substitution of Sr for Ca is accompanied by a transfer of spectral weight between sliding modes associated with different chain-ladder dynamics. The observation is consistent with a transfer of hole charges from the chains to the ladders and modification of the sublattice dimensions following Ca substitution. The results are discussed in context to the significance of low-lying vibrational dynamics and electron-phonon coupling in the superconducting state of certain quasiperiodic systems.