Local causality consists of 2 premises:
a) Causality, the idea that effects have causes that precede them. Ie some set of conditions makes for a subsequent unique outcome.
b) Locality, the idea that only conditions that are "nearby" can be part of determining the unique outcome.
Local Causality can be violated if either of those assumptions are unwarranted. For example, Locality is violated if there are long-range forces (currently unknown) which propagate faster than c. Causality is violated is time is symmetric (the future affects the past) or if there are not unique outcomes (Many Worlds).