The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic BK□, which lacks a primitive possibility operator ◊, is definitionally equivalent with the logic BK, which has both and ◊ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with BK□ without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic BK□ × BK□ over the non-modal vocabulary of MBL. On the way from BK□ to MBL, the Fischer Servi-style modal logic BKFS is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and BKFS is shown to be characterized by the class of all models for BK□ × BK□. Moreover, BKFS is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for BK□ x BK□. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that BKFS and BK□ × BK□ are weakly definitionally equivalent.