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Tri-stable piezoelectric vibration energy harvester (TPVEH) with asymmetric potential wells has lower potential difference and more unstable potential wells than the symmetric counterpart, which may benefit to improve the energy harvesting performance. Although some progress has been made, there is still few research on the asymmetric TPVEH compared to the symmetric counterparts. The bifurcation mechanism of equilibrium solutions, the transition mechanism of multi-stability state and the nonlinear dynamics of the asymmetric TPVEH are different from those of the symmetric one, and have not been well explored at present. This work presents an asymmetric TPVEH which is composed of a piezoelectric linear oscillator (PLO) and a magnet–mass–spring oscillator (MMSO) to theoretically and experimentally investigate the bifurcations, the transition mechanism and the nonlinear dynamics. A general nonlinear dynamic model of the asymmetric TPVEH is derived and its equilibrium solutions are calculated to reveal the bifurcation and transition mechanism. The steady-state response solutions and their stability analysis are successively deduced to theoretically investigate the nonlinear dynamics and the enhancing mechanism of energy harvesting. Experiments are consequently performed to validate the theoretical results. The results show that the MMSO can not only change the distribution position of the stable and unstable equilibrium points, but also change the potential barriers height by adjusting the horizontal interval in real-time. In addition, suitable spring stiffness of the MMSO benefits to enhance the dynamic response outputs and frequency bandwidth. Compared to the symmetric TPVEH, the asymmetric TPVEH can access the interwell oscillations more easily and generate larger response outputs over a wide frequency range under lower excitation level. For the suitable spring stiffness of 1000 N/m, the maximum voltage, power obtained by experiments and the acceleration threshold for snap-through of the asymmetric TPVEH are 7 V, 4.8 × 10−5 W and 8 m/s2, respectively, which are far better than the symmetric TPVEH, whose corresponding values are 3 V, 0.8 × 10−5 W and 18 m/s2, respectively. • An asymmetric TPVEH with a MMSO is proposed for energy harvesting enhancement. • The bifurcations and nonlinear dynamics are comprehensively investigated. • The MMSO can adjust the magnet distance to generate asymmetric potential wells. • The MMSO with suitable spring stiffness benefits the energy harvesting performances. • The asymmetric TPVEH has significant advantages over the symmetric counterpart in energy harvesting. [ABSTRACT FROM AUTHOR]