In the Block LMS algorithm, the input is partitioned into nonoverlapping blocks of size $L$ and the filter coefficients are updated once every $L$ samples. Would convergence improve if we used more frequent filter coefficient updates by using overlapping blocks? Having more "descent steps" per unit time would seem to imply faster convergence, but the lack of mention of this in any literature suggests otherwise.
There is no hard rule regarding convergence speed of the block-LMS vs sample-by-sample LMS. It really depends on the scenario. On top of my head is the following two (stationary) scenarios:
A very noisy scenario, where a single estimate of the gradient is not enough. In this case, the block-LMS has better gradient estimates and would usually result in faster convergence.
A clean scenario. In this one, the sample-by-sample LMS will yield a gradient estimate that is almost as good as the block-based LMS. Then, the sample-by-sample LMS will show a better convergence behavior.
Other examples can be constructed using stationary/non-stationary distinction.