Firstly, it is introduced that the concepts of G-almost periodic point and G-sequence shadowing property. Then, we discuss the dynamical relationship between sequence map {g_k}^∞_k=1 and limit map g under G-strongly uniform convergence of topological group action. We can get that (1) Let sequence map {g_k}^∞_k=1 be G-strongly uniform converge to the map g where g is equicontinuous and the point sequence {y_k}^∞_k=1 be the G-almost periodic point of sequence map {g_k}^∞_k=1. If lim_{k → ∞} y_k = y, then the point y is an G-almost periodic point of the map g; (2) If sequence map {g_k}^∞_k=1 are G-strongly uniform converge to the map g where g is equicontinuous, then limsup AP_G(g_k) ⊂ AP_G(g); (3) Let sequence map {g_k}^∞_k=1 be G-strongly uniform converge to the map g. If every map g_k has G-fine sequence shadowing property, the map g has G-sequence shadowing property. These results generalize the corresponding results given in Ji and Zhang [1] and make up for the lack of theory under G-strongly uniform convergence of group action.