Background: Accurate deformable image registration (DIR) is required for contour propagation and dose accumulation in MR-guided adaptive radiotherapy (MRgART). This study trained and evaluated a deep learning DIR method for domain invariant MR-MR registration. Methods: A progressively refined registration and segmentation (ProRSeg) method was trained with 262 pairs of 3T MR simulation scans from prostate cancer patients using weighted segmentation consistency loss. ProRSeg was tested on same- (58 pairs), cross- (72 1.5T MR Linac pairs), and mixed-domain (42 MRSim-MRL pairs) datasets for contour propagation accuracy of clinical target volume (CTV), bladder, and rectum. Dose accumulation was performed for 42 patients undergoing 5-fraction MRgART. Results: ProRSeg demonstrated generalization for bladder with similar Dice Similarity Coefficients across domains (0.88, 0.87, 0.86). For rectum and CTV, performance was domain-dependent with higher accuracy on cross-domain MRL dataset (DSCs 0.89) versus same-domain data. The model's strong cross-domain performance prompted us to study the feasibility of using it for dose accumulation. Dose accumulation showed 83.3% of patients met CTV coverage (D95 >= 40.0 Gy) and bladder sparing (D50 <= 20.0 Gy) constraints. All patients achieved minimum mean target dose (>40.4 Gy), but only 9.5% remained under upper limit (<42.0 Gy). Conclusions: ProRSeg showed reasonable multi-domain MR-MR registration performance for prostate cancer patients with preliminary feasibility for evaluating treatment compliance to clinical constraints.