At the Heidelberg Ion-Beam Therapy Center, patient irradiation with scanned proton and carbon ion beams is verified by offline positron emission tomography (PET) imaging: the β+-activity measured within the patient is compared to a prediction calculated on the basis of the treatment planning data in order to identify potential delivery errors. Currently, this monitoring technique is limited to the treatment of static target structures. However, intra-fractional organ motion imposes considerable additional challenges to scanned ion beam radiotherapy. In this work, the feasibility and potential of time-resolved (4D) offline PET-based treatment verification with a commercial full-ring PET/CT (x-ray computed tomography) device are investigated for the first time, based on an experimental campaign with moving phantoms. Motion was monitored during the gated beam delivery as well as the subsequent PET acquisition and was taken into account in the corresponding 4D Monte-Carlo simulations and data evaluation. Under the given experimental conditions, millimeter agreement between the prediction and measurement was found. Dosimetric consequences due to the phantom motion could be reliably identified. The agreement between PET measurement and prediction in the presence of motion was found to be similar as in static reference measurements, thus demonstrating the potential of 4D PET-based treatment verification for future clinical applications.