As a new heuristic algorithm, tumbleweed algorithm (TA) has many advanced versions. However, finding the dynamic balance between the exploration and exploitation of it is still a problem to be considered. This paper proposes an excellent multi-stage adaptive perturbation tumbleweed algorithm (MSAP-TA) to further improve the performance of TA. Firstly, a multi-population regional initialization and co-evolution mechanism is set up to enhance the global exploration capability of the algorithm. Secondly, a phased dynamic scheduling strategy is introduced. Different search and perturbation methods are adopted in different stages to realize the progressive convergence. Finally, an elite trajectory archive and a stagnation adjustment method are applied to avoid the algorithm from falling into the local optimum. In the end, the MSAP-TA is compared with five other heuristic algorithms on CEC2020 and CEC2013. The experimental results show its strong competitiveness and application value.