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Topological edge states offer a powerful platform for understanding and manipulating waves, encompassing classical electromagnetic and acoustic wave systems and condensed matter systems. Though topological states have been recently extended from wave to thermal diffusion systems, their realization is still confined to pure thermal conduction, unable to provide a general discussion as lacking access to thermal advection. Here, we attempt to extend the concept of topological edge states from diffusion to diffusion-advection dynamics and reveal the dynamically adjustable counterpart topological edge states. The varying ratios of inter- to intra-coupling advection coefficients can profoundly impact the time evolution of thermal systems, precipitating the emergence of distinct topological states, which cannot happen in a pure conduction system. The diffusion-advection systems are non-Hermitian with eigenvalues consisting of the imaginary component corresponding to diffusion rate and the real component corresponding to advection rate. Upon deriving the governing Hamiltonian of a thermal lattice, we observe the emergence of topological edge states in the gap of bulk states. Additionally, we present a comparative analysis that underscores the unique thermal properties of bulk and edge states enabled by the thermal advection term. Our work paves the way for the exploration of topological states within thermal diffusion-advection systems and potentially provide distinct paradigms for efficient thermal management and thermal protection.