Abstract. In this paper, we study the Neumann boundary value problem of the Yang--Mills $\alpha$-flow over a 4-dimensional compact Riemannian manifold with boundary. We establish the short-time existence of the Yang--Mills $\alpha$-flow in the framework of functional analysis and derive long-time existence and convergence results of classical solutions to the Yang--Mills $\alpha$-flow, provided that the $\alpha$-energy of the initial connection is below some threshold. We also prove the validity of the boundary version of small energy estimates, removal of isolated singularities, and the gap theorem. These results lead to the bubbling convergence of a sequence of Yang--Mills $\alpha$-connections, and as an application, we demonstrate the existence of non-trivial Yang--Mills connections with Neumann boundary.