Spatial-temporal data, fundamental to many intelligent applications, reveals dependencies indicating causal links between present measurements at specific locations and historical data at the same or other locations. Within this context, adaptive spatial-temporal graph neural networks (ASTGNNs) have emerged as valuable tools for modelling these dependencies, especially through a data-driven approach rather than pre-defined spatial graphs. While this approach offers higher accuracy, it presents increased computational demands. Addressing this challenge, this paper delves into the concept of localisation within ASTGNNs, introducing an innovative perspective that spatial dependencies should be dynamically evolving over time. We introduce \textit{DynAGS}, a localised ASTGNN framework aimed at maximising efficiency and accuracy in distributed deployment. This framework integrates dynamic localisation, time-evolving spatial graphs, and personalised localisation, all orchestrated around the Dynamic Graph Generator, a light-weighted central module leveraging cross attention. The central module can integrate historical information in a node-independent manner to enhance the feature representation of nodes at the current moment. This improved feature representation is then used to generate a dynamic sparse graph without the need for costly data exchanges, and it supports personalised localisation. Performance assessments across two core ASTGNN architectures and nine real-world datasets from various applications reveal that \textit{DynAGS} outshines current benchmarks, underscoring that the dynamic modelling of spatial dependencies can drastically improve model expressibility, flexibility, and system efficiency, especially in distributed settings.