The original source that I often return to (even before I look in my book) is Wendland's 2005 book "Scattered Data Approximation". But that book is really tough to read, even for me. Lots of tough math. Fasshauer's 2007 book "Meshfree Approximation Methods in Matlab" is much easier to read, but references Wendland's book for most of the heavy lifting.
There is a newer branch of study on convergence dealing with what are called "sampling inequalities". In many ways, the math behind these is even worse as it usually requires a bunch of polynomial theory, but the results are more easily accessible. In particular, Christian Rieger and Barbara Wolmuth have outstanding content on this which is readily available on the web. In particular, I was able to find Christian's PhD thesis (https://www.deutsche-digitale-bibliothek.de/binary/JOWXLCGSV...) which provides a good journey from the start to actual results.
Sorry I can't provide a cleaner statement about this. Maybe the most basic point I can make about convergence here would be to reference the early part of Christian's thesis where he alludes to the theorem that says the quality of an interpolant sort of looks like:
error of interpolant = O(h^{k-d})
That is a gross simplification, but gives the gist of the result. h is the "fill distance" (or grid width for structured data) k is the smoothness of the kernel and d is the dimension of the data. This indicates why smooth kernels (large k) are needed to get the convergence for high d.