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A method for simulating spot-scanned delivery to a moving tumour was developed which uses patient-specific image and plan data. The magnitude of interplay effects was investigated for two patient cases under different fractionation and respiratory motion variation scenarios. The use of volumetric rescanning for motion mitigation was also investigated. For different beam arrangements, interplay effects lead to severely distorted dose distributions for a single fraction delivery. Baseline shift variations for single fraction delivery reduced the dose to the clinical target volume (CTV) by up to 14.1 Gy. Fractionated delivery significantly reduced interplay effects; however, local overdosage of 12.3% compared to the statically delivered dose remained for breathing period variations. Variations of the tumour baseline position and respiratory period were found to have the largest influence on target inhomogeneity; these effects were reduced with fractionation. Volumetric rescanning improved the dose homogeneity. For the CTV, underdosage was improved by up to 34% in the CTV and overdosage to the lung was reduced by 6%. Our results confirm that rescanning potentially increases the dose homogeneity; however, it might not sufficiently compensate motion-induced dose distortions. Other motion mitigation techniques may be required to additionally treat lung tumours with scanned proton beams.