This paper introduces the concept of time-varying Koopman operator to predict the flow of a nonlinear dynamical system. The Koopman operator provides a linear prediction model for nonlinear systems in a lifted space of infinite dimension. An extension of time-invariant subspace realization methods known as the time-varying Eigensystem Realization Algorithm (TVERA) in conjunction with the time-varying Observer Kalman Identification Algorithm (TVOKID) are used to derive a finite dimensional approximation of the infinite dimensional Koopman operator at each time step. An isomorphic coordinate transformations is defined to convert different system realizations from different sets of experiments into a common frame for state propagation and to extract dynamical features in the lifted space defined by the eigenvalues of the Koopman operator. Two benchmark numerical examples are considered to demonstrate the capability of the proposed approach.