This paper investigates the idea of reasoning, in a local (or contextual) way, under prioritized and possibly inconsistent knowledge bases. Priorities are not supposed to be given globally between all the beliefs in the knowledge base, but locally inside sets of pieces of information responsible for inconsistencies. This local stratification offers more flexibility for representing priorities between beliefs. Given this local ordering, we discuss five basic definitions of influence relations between conflicts. These elementary notions of influence between two conflicts A (...) and B exhaustively explore the situations where solving A leads to solve B. Then we propose natural approaches to restore the coherence of a knowledge base on the basis of influence relations between locally-stratified conflicts. (shrink)