Information-Centric Networking (ICN) originally innovated for efficient data distribution, is currently discussed to be applied to edge computing environment. In this paper, we focus on a more flexible context, in-network computing, which is enabled by ICN architecture. In ICN-based in-network computing, a function chaining (routing) method for chaining multiple functions located at different routers widely distributed in the network is required. Our proposal is a twofold approach, On-demand Routing for Responsive Route (OR 3 ) and Route Records (RR). OR 3 efficiently chains data and multiple functions compared with an existing routing method. RR reactively stores routing information to reduce communication/computing overhead. In this paper, we conducted a mathematical analytics in order to verify the correctness of the proposed routing algorithm. Moreover, we investigate applicabilities of OR 3 /RR to an edge computing context in the future Beyond 5G/6G era, in which rich computing resources are provided by mobile nodes thanks to the cutting-edge mobile device technologies. In the mobile environments, the optimum from viewpoint of "routing" is largely different from the stable wired environment. We address this challenging issue and newly propose protocol enhancements for OR 3 by considering node mobility. Evaluation results reveal that mobilityenhanced OR 3 can discover stable paths for function chaining to enable more reliable ICN-based in-network computing under the highly-dynamic network environment.