The Distributed Adaptive Signal Fusion (DASF) framework is a meta-algorithm for computing data-driven spatial filters in a distributed sensing platform with limited bandwidth and computational resources, such as a wireless sensor network. The convergence and optimality of the DASF algorithm has been extensively studied under the assumption that an exact, but possibly impractical solver for the local optimization problem at each updating node is available. In this work, we provide convergence and optimality results for the DASF framework when used with an inexact, finite-time solver such as (proximal) gradient descent or Newton's method. We provide sufficient conditions that the solver should satisfy in order to guarantee convergence of the resulting algorithm, and a lower bound for the convergence rate. We also provide numerical simulations to validate these theoretical results.