We investigate a new class of function spaces called variable bounded variation spaces of higher order by generalizing the classical notion of bounded variation. Our approach incorporates concepts from higher-order variations, bounded variation in the sense of generalized norms, and variations defined through modular function spaces. We focus on analysing the approximation properties of convolution-type nonlinear integral operators within these newly defined function spaces. In particular, we study these operators on variable bounded variation spaces of a fixed order in the sense of higher variations. Several approximation results are obtained using regular summability methods, highlighting the effectiveness of the proposed approach in improving convergence behaviour.