Why does diagonal loading of a covariance matrix make an adaptive beamformer more robust in the case of a perturbed array?
I had worked on an array processing problem where I had used diagonal loading of the measurement covariance matrix. But I had used diagonal loading as a solution to what I thought was a numerical issue with my eigen values being too small. Since I had to invert the covariance matrix with small eigen values, I had numerical issues with the result. 
Diagonal loading of the covariance matrix helped me get rid of the issue as the eigen values were now bounded by the diagonal loading value.

I am not sure, but one place where diagonal loading helps is in numerical stability of inverting the covariance matrix which is most likely encountered in adaptive beamforming.