In this paper we investigate a refinement of minimal abnormal and reliable models in modal adaptive logics. Starting from lower limit logics below S5, we introduce a new flavour of localised adaptive consequence based on minimally abnormal or reliable point generated submodels. The design of a proof-system for such a localised consequence, asks for a labelling mechanism allowing the derivation of formulas at a point in a frame. A labelled dynamic proof-format is therefore elaborated in the second part of the paper.