This work presents a novel approach for \textit{bearing rigidity} analysis and control in multi-robot networks with sensing constraints and dynamic topology. By decomposing the system's framework into \textit{subframeworks}, we express bearing rigidity -- a global property -- as a set of \textit{local} properties, with rigidity eigenvalues serving as natural \textit{local rigidity measures}. We propose a decentralized gradient-based controller to execute mission-specific commands using only bearing measurements. The controller preserves bearing rigidity by keeping the rigidity eigenvalues above a threshold, using only information exchanged within subframeworks. Simulations evaluate the scheme's effectiveness, underscoring its scalability and practicality.