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In mesoscopic physics, the application of a time-periodic drive leads to novel transport behaviour, which is absent in the static regimes. Here we consider a quantum pumping protocol, such that the quasiparticles of Weyl/multi-Weyl and nodal-line semimetals are subjected to a time-periodic rectangular potential well. The presence of an oscillating potential of frequency $ω$ creates equispaced Floquet side-bands with spacing $\hbar ω$. As a result, a Fano resonance is observed when the difference in the Fermi energy (i.e., the energy of the incident quasiparticle), and the energy of one of the (quasi)bound state levels of the well, coincides with the energy of an integer number of photons (each carrying energy quantum $\hbar ω$, equal to the side-band spacing). Using the Floquet theory and the scattering matrix approach, in the zero-temperature non-adiabatic pumping limit, we find characteristic Fano resonance patterns in the transmission coefficients, which depend on the nature of the dispersion. The inflection points in the pumped shot noise spectra also serve as a proxy for the corresponding Fano resonances. Therefore, we also numerically evaluate the pumped shot noise. Finally, we correlate the existence of the Fano resonance points to the (quasi)bound states of the well, by explicitly calculating the bound states of the static well (which are a subset of the bound states of the driven system). Since we consider semimetals with anisotropic dispersions, all the features observed depend on the orientation of the potential well. We believe that our results will serve as a guide for future experiments investigating quantum transmission in nonequilibrium settings.