Why does block LMS have the same performance as LMS?
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.)