Extending local kinematic studies to earlier cosmic times is valuable to understand how galaxies evolve in relation to their dark matter haloes. In a series of papers on lensed kinematics, we seek to combine the sensitivity of 3D forward modelling to low signal-to-noise ratio outskirts with the enhanced spatial resolution provided by cluster lensing. In this first paper, we (i) present and validate our methodology, which directly constrains the source parameters by incorporating lensing deflections into the GalPaK3D forward-modelling algorithm, and (ii) investigate the evolution of the stellar-mass and baryonic-mass Tully-Fisher relations (sTFR and bTFR) since z~1 as a demonstration. We define a robust sample of strongly lensed star-forming galaxies (SFGs) from the MUSE Lensing Cluster survey, spanning magnifications mu=1.4-12.4 and stellar masses M*=10^8.1^-10^10.3^M__{sun}_. Using a series of mock galaxies representative of our sample, we find that our method is significantly more reliable at recovering morpho-kinematic properties than approaches that ignore differential magnification, even for relatively modest magnifications (mu<6). Restricting the analysis to 95 rotationally supported SFGs with well-constrained velocities, we find a significant evolution of the sTFR zero-point ({Delta}b^sTFR^=-0.42^+0.05^_-0.05_dex in stellar mass) but no detectable evolution of the bTFR zero-point ({Delta}b^bTFR^=0.00^+0.06^_-0.06_dex in baryonic mass) relative to z~0. Our results are consistent with a mild evolution of the stellar-to-halo mass ratio and support the view that the sTFR has evolved only weakly over the past ~8Gyr, aside from shifts driven by the redshift dependence of halo-defining quantities such as the critical density and overdensity. The absence of detectable evolution in the bTFR zero-point suggests that the increasing contribution of cold gas mass at higher redshift fully compensates the evolution observed in the stellar component alone.