Large spatial datasets often exhibit features that vary at different scales as well as at different locations. To model random fields whose variability changes at differing scales we use multiscale kernel convolution models. These models rely on nested grids of knots at different resolutions. Thus, lower order terms capture large scale features, while high order terms capture small scale ones. In this talk we consider two approaches to fitting multi-resolution models with space-varying characteristics. In the first approach, to accommodate the space-varying nature of the variability, we consider priors for the coefficients of the kernel expansion that are structured to provide increasing shrinkage as the resolution grows. Moreover, a tree shrinkage prior auto-tunes the degree of resolution necessary to model a subregion in the domain. In addition, compactly supported kernel functions allow local updating of the model parameters which achieves massive scalability by suitable parallelization. As an alternative, we develop an approach that relies on knot selection, rather than shrinkage, to achieve parsimony, and discuss how this induces a field with spatially varying resolution. We extend shotgun stochastic search to the multi resolution model setting, and demonstrate that this method is computationally competitive and produces excellent fit to both synthetic and real dataset.