Abstract Objective. Respiratory motion of lung tumours and adjacent structures is challenging for radiotherapy. Online MR-imaging cannot currently provide real-time volumetric information of the moving patient anatomy, therefore limiting precise dose delivery, delivered dose reconstruction, and downstream adaptation methods. Approach. We tailor a respiratory motion modelling framework towards an MR-Linac workflow to estimate the time-resolved 4D motion from real-time data. We develop a multi-slice acquisition scheme which acquires thick, overlapping 2D motion-slices in different locations and orientations, interleaved with 2D surrogate-slices from a fixed location. The framework fits a motion model directly to the input data without the need for sorting or binning to account for inter- and intra-cycle variation of the breathing motion. The framework alternates between model fitting and motion-compensated super-resolution image reconstruction to recover a high-quality motion-free image and a motion model. The fitted model can then estimate the 4D motion from 2D surrogate-slices. The framework is applied to four simulated anthropomorphic datasets and evaluated against known ground truth anatomy and motion. Clinical applicability is demonstrated by applying our framework to eight datasets acquired on an MR-Linac from four lung cancer patients. Main results. The framework accurately reconstructs high-quality motion-compensated 3D images with 2 mm³ isotropic voxels. For the simulated case with the largest target motion, the motion model achieved a mean deformation field error of 1.13 mm. For the patient cases residual error registrations estimate the model error to be 1.07 mm (1.64 mm), 0.91 mm (1.32 mm), and 0.88 mm (1.33 mm) in superior-inferior, anterior-posterior, and left-right directions respectively for the building (application) data. Significance. The motion modelling framework estimates the patient motion with high accuracy and accurately reconstructs the anatomy. The image acquisition scheme can be flexibly integrated into an MR-Linac workflow whilst maintaining the capability of online motion-management strategies based on cine imaging such as target tracking and/or gating.