We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Itô’s formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler–Maruyama (EM) discretization for multi-dimensional Ornstein–Uhlenbeck stable process and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance.