Why does block LMS have the same performance as LMS? 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?
here is a short and little (a.k.a. "consise but terse") derivation of the LMS and normalized LMS adaptive filter.

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.)