The block LMS and conventional LMS have the same convergence rate and the same misadjustment. I am having trouble wrapping my head around this. The block LMS uses a more accurate estimate of the gradient vector at each iteration. Conceptually, why does a better gradient estimate have no benefit on the descent?
once the LMS has converged on a reasonably stable equilibrium for $h_n[k]$, they won't move around that much. then it doesn't matter so much how long the block is. and the only difference between block-LMS and the plain-old ordinary LMS is the block size. (the block size for the latter is 1.)