This paper studies the convergence conditions and properties of the distributed adaptive signal fusion (DASF) algorithm, the framework itself having been introduced in a `Part I' companion paper. The DASF algorithm can be used to solve linear signal and feature fusion optimization problems in a distributed fashion, and is in particular well-suited for solving spatial filtering optimization problems encountered in wireless sensor networks. The convergence conditions and results are provided along with rigorous proofs and analyses, as well as various example problems to which they apply. Additionally, we describe procedures that can be added to the DASF algorithm to ensure convergence in specific cases where some of the technical convergence conditions are not satisfied.