The preferential conditional logic $\mathbb{PCL}$, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalise Lewis' sphere models for counterfactual logics, is proposed. Soundness and completeness of $\mathbb{PCL}$ and its extensions with respect to this class of models are proved directly. Labelled sequent calculi for all logics of the family are then introduced. The cal-culi are modular and have standard proof-theoretical properties, the most important of which is admissibility of cut, that entails a syntactic proof of completeness of the calculi. By adopting a general strategy, root-first proof search terminates, thereby providing a decision procedure for $\mathbb{PCL}$ and its extensions. Finally, the semantic completeness of the calculi is established: from a finite branch in a failed proof attempt it is possible to extract a finite countermodel of the root sequent. The latter result gives a constructive proof of the finite model property of all the logics considered.