Owen Chadwick devotes a chapter of The Reformation to “the decline of ecclesiastical power.” He reviews the effects of the Reformation on excommunication, the benefit of clergy, church property and the power of the local parish church. Sanctuary was one of the medieval institutions ended by the Reformation.
Sanctuary rights were already under suspicion in the medieval period: “In the later Middle Ages [sanctuary] became a way of evading justice. In some notorious cases, like the church of St Martin-le-Grand in London in the earthly fifteenth century, a group of men living a public sanctuary became little more than a band of brigands issuing forth from an immune fastness.” Not everyone was a criminal: “nearly 500 people took refuge [at St John at Beverly] in the sixty years before 1538” (380).
Chadwick estimates that “most governments of the later Middle Ages, including the Popes, attempted to abridge the right.” In the sixteenth century, the “more powerful states” in Europe suppressed what they saw as “anarchism.” Henry VIII “excluded from sanctuary persons guilty of murder, rape, highway robbery, burglary, arson, or sacrilege.” Every Protestant “country abolished the right during the course of their reformation, some more slowly than others.” Roman Catholic countries curtailed the right, much as Henry VIII had, but “it became a matter of principle to maintain it, chiefly as a sign of ecclesiastical independence, despite the manifest inconveniences which it caused.” A semblance of sanctuary is still in force in Italy, where “secular officials shall not exercise their functions in a church without giving previous notice to the ecclesiastical authorities” (380-1).
Chadwick claims that sanctuary “was appropriate to a simpler world,” and was undone by “the efficiency of public justice” (381). That's one way to put it. It might also be described as a curtailment of the independence of the church in the face of an monopolizing state.