Graph signal processing (GSP) provides a framework for analyzing data on irregular domains, with applications in neuroscience, finance, chemistry, and social sciences. Classical GSP primarily models symmetric relationships using undirected graphs, yet many real-world systems exhibit asymmetric interactions, motivating extensions to directed graphs. Central to directed GSP is the graph shift operator, typically defined via the directed graph Laplacian. Building on the well-known link between the undirected graph Laplacian and the diffusion operator, we establish a correspondence between the directed graph Laplacian and the diffusion-advection operator. This perspective opens new avenues for addressing crucial points such as frequency ordering, smoothness definition, and the design of spectral and graph filters. Specifically, we introduce two new orderings of frequencies based on the modulus and argument of the eigenvalues, naturally leading to new definitions of smoothness. Then we present two kernels reflecting diffusive and advective processes, namely the heat and transport kernels, respectively. Finally, we propose novel graph filters obtained by composing diffusive and advective parts, which approximate ideal spectral filters accurately and characterize the evolution of graph signals in richer ways. All aforementioned developments are illustrated on both synthetic and real graphs, including an application to temperature graph and signals.