Block LMS with overlapping blocks
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.