A novel computational framework for reconstructing spatial and temporal profiles of moving acoustic sources from wave responses measured at sparsely distributes sensors is introduced in this paper. This method can be applied to a broad range of acoustic-source inversion (ASI) problems for heterogeneous, complex-shaped coupled dynamic systems. The finite element method (FEM) is used to obtain wave response solutions due guessed moving sources. An adjoint-gradient based optimization technique iteratively improves the guesses so that the guessed moving sources converge on the actual moving sources. To reconstruct acoustic source profiles without a-priori knowledge of sources, we will employ high-resolution discretization of source functions in space and time. Because of such dense discretization, the order of magnitude of number of inversion parameters could range from millions to billions. Numerical experiments prove the robustness of this method by reconstructing spatial and temporal profiles of multiple dynamic moving body forces in a one-dimensional heterogeneous solid bar. The sources create stress waves propagating through the bar. The guessed source functions are spatially discretized by using linear shape functions with an element size of 1m at discrete times with a time step of 0.001s. Thus, the total number of control parameters in this example is 100,000 (i.e., 100 (in space) by 1000 (in time)). The convergence toward the target in the numerical examples is excellent, reconstructing the spatial and temporal footprints of the sources.