This paper studies the fixed point finding problem for a global operator over a directed and unbalanced multi-agent network, where the global quasi-nonexpansive operator is sum separable and composed of a family of local operators. In this problem, each local operator is privately known to only each individual agent, and all local operators are assumed to be Lipschitz continuous. To deal with this problem, a distributed (or decentralized) algorithm, called Distributed quasi-averaged Operator Tracking algorithm (DOT), is proposed and rigorously analyzed, and it is shown that the algorithm can converge to a fixed point of the global operator at a linear (or exponential) rate under a bounded linear regularity condition, which is strictly weaker than the function’s strong convexity in convex optimization. To validate the proposed algorithm, a numerical example is provided finally.