Current integrated photonics typically utilizes the dynamical phase associated resonant on-chip components, leading to bandwidth-limited devices with perturbation-sensitive performance. Multi-mode geometric phase matrix, arising from the non-Abelian holonomy which is a non-resonant and global effect, has been recently introduced to integrated photonics for the design of broadband and robust on-chip photonic devices. Achieving reconfigurable on-chip non-Abelian photonic devices is a crucial step towards practical applications, which however remains elusive. Here, we propose a universal approach by employing the thermo-optic effect to tune the system’s Hamiltonian and thus the holonomy induced geometric phase matrix. We implement this concept in double-layered polymer integrated platforms, experimentally demonstrating a four-mode non-Abelian braiding device comprising six sets of tunable two-mode braiding building blocks. Through modulations, the device can be reconfigured to generate up to 24 unitary matrices belonging to the braid group B₄. Our work paves the way for non-Abelian integrated photonics towards abundant applications.