We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using some recent dimension formulas.