The flow through tandem square cylinders was investigated at a Reynolds number of 100 for oscillation amplitudes A = 0.1D to 0.7D and gaps L = 2.0D, 5.0D, and 6.0D, where D is the width of the cylinders. A moving reference frame method combined with the spectral/hp element method was employed to simulate the two-dimensional flow in the lock-in regime. Fluid forces, vorticity fields, power spectrum density, and pressure distribution were first investigated. Since surface pressure is directly connected with fluid forces, pressure and velocity field were synchronously analyzed by employing optimal dynamic mode decomposition. An underlying link between fluid forces and coherence modes was then uncovered. The results reveal that the move-induced forces and flow structures strongly depend on gaps and amplitudes in the lock-in regime. With respect to the dynamic mode decomposition analysis, odd-order modes contribute to lift forces, while even-order modes result in drag forces. The flow structures are dominated by at most three modes; as the amplitude increases, the high-order mode energy increases, coinciding with corresponding power spectrum density results of forces. Typical 2S, 2P, and C(2S) wakes were observed for various gaps and two representative amplitudes (A/D = 0 and 0.7), and their dominant modes show distinctive differences that lead to different local pressure shapes on the cylinders. It is the combined effects of local mode shape and global mode energy that account for the change in fluid forces for various gaps and two oscillating amplitudes.