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Abdominal organ segmentation from CT and MRI is an essential prerequisite for surgical planning and computer-aided navigation systems. It is challenging due to the high variability in the shape, size, and position of abdominal organs. Three-dimensional numeric representations of abdominal shapes with point-wise correspondence to a template are further important for quantitative and statistical analyses thereof. Recently, template-based surface extraction methods have shown promising advances for direct mesh reconstruction from volumetric scans. However, the generalization of these deep learning-based approaches to different organs and datasets, a crucial property for deployment in clinical environments, has not yet been assessed. We close this gap and employ template-based mesh reconstruction methods for joint liver, kidney, pancreas, and spleen segmentation. Our experiments on manually annotated CT and MRI data reveal limited generalization capabilities of previous methods to organs of different geometry and weak performance on small datasets. We alleviate these issues with a novel deep diffeomorphic mesh-deformation architecture and an improved training scheme. The resulting method, UNetFlow, generalizes well to all four organs and can be easily fine-tuned on new data. Moreover, we propose a simple registration-based post-processing that aligns voxel and mesh outputs to boost segmentation accuracy.