Heritable jurisdiction could be expensive. George Heriot's School in Edinburgh ended up with the right, by virtue of the lands they owned (previously held by such as Logan of Restalrig). At some point in the early 18th century the trustees of Heriot's actually got stuck with the responsibility of organizing a capital trial, or were about to (the case of Nicol Muschat? I'm not sure) and petitioned the city council to take the whole business off their hands as it was eating time and money they needed for educational purposes.

You'd think a school could take useful advantage of being able to sentence people to death (talk back to the teacher and get strung up in the quadrangle) but since they couldn't make any laws and had to involve lawyers, they had all the hassle and none of the fun.