In this paper, we unify popular non-rigid registration methods for point sets and surfaces under our general framework, GiNGR. GiNGR builds upon Gaussian Process Morphable Models (GPMM) and hence separates modeling the deformation prior from model adaptation for registration. In addition, it provides explainable hyperparameters, multi-resolution registration, trivial inclusion of expert annotation, and the ability to use and combine analytical and statistical deformation priors. But more importantly, the reformulation allows for a direct comparison of registration methods. Instead of using a general solver in the optimization step, we show how Gaussian process regression (GPR) iteratively can warp a reference onto a target, leading to smooth deformations following the prior for any dense, sparse, or partial estimated correspondences in a principled way. We show how the popular CPD and ICP algorithms can be directly explained with GiNGR. Furthermore, we show how existing algorithms in the GiNGR framework can perform probabilistic registration to obtain a distribution of different registrations instead of a single best registration. This can be used to analyze the uncertainty e.g. when registering partial observations. GiNGR is publicly available and fully modular to allow for domain-specific prior construction.