Incorporating graphs in the analysis of multivariate signals is becoming a standard way to understand the interdependency of activity recorded at different sites. The new research frontier in this direction includes the important problem of how to assess dynamic changes of signal activity. We address this problem in a novel way by defining the graph-variate signal alongside methods for its analysis. Essentially, graph-variate signal analysis leverages graphs of reliable connectivity information to filter instantaneous bivariate functions of the multivariate signal. This opens up a new and robust approach to analyse joint signal and network dynamics at sample resolution. Furthermore, our method can be formulated as instantaneous networks on which standard network analysis can be implemented. When graph connectivity is estimated from the multivariate signal itself, the appropriate consideration of instantaneous graph signal functions allows for a novel dynamic connectivity measure-- graphvariate dynamic (GVD) connectivity-- which is robust to spurious short-term dependencies. Particularly, we present appropriate functions for three pertinent connectivity metrics-- correlation, coherence and the phase-lag index. We show that our approach can determine signals with a single correlated couple against wholly uncorrelated data of up to 128 nodes in signal size (1 out of 8128 weighted edges). GVD connectivity is also shown to be more robust than i) other GSP approaches at detecting a randomly traveling spheroid on a 3D grid and ii) standard dynamic connectivity in determining differences in EEG restingstate and task-related activity. We also demonstrate its use in revealing hidden depth correlations from geophysical gamma ray data. We expect that the methods and framework presented will provide new approaches to data analysis in a variety of applied settings.