The paper provides an analysis of dynamic behavior of peristaltic transport of non-Newtonian fluid in a nonuniform diverging channel with various waveforms. The canonical object of the study is the bifurcation techniques of the physical parameters, from which information on the dynamic response of peristaltic flow can be gained. Special attention is paid to the interaction between local and global dynamics through a nonuniform channel with different wall waveforms, which is shown to generate a range of creative behaviors, involving heteroclinic and homoclinic connections to saddle stagnation points. These closed invariant curves form a novel phenomenon involving different flow scenarios in a finite region, without the need for varying parameters. The bifurcation analytical study is complimented by numerical computations, both of which are used to highlight the impacts predicted on flow parameters, such as Grashof, solute Grashof, heat source/sink, and thermal radiation parameters. We show that properly accounting for the interaction between invariant sets, multiple stagnation points, and streamline patterns leads to unprecedented levels of flow control characteristics. We also compare the bifurcation behaviors of peristaltic transport through uniform and nonuniform channel under different waveforms that will be useful for the topologies controlling stream flow with complex shape.