In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation in all ranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the small Knudsen number regime. The first APNN for linear transport equation is based on even-odd decomposition, which relaxes the stringent conservation prerequisites while concurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundary conditions is crucial for this network. For the Boltzmann-BGK equation, the APNN incorporates the conservation of mass, momentum, and total energy into the APNN framework as well as exact boundary conditions. A notable finding of this study is that approximating the zeroth, first, and second moments—which govern the conservation of density, momentum, and energy for the Boltzmann-BGK equation, is simpler than the distribution itself. Another interesting phenomenon observed in the training process is that the convergence of density is swifter than that of momentum and energy. Finally, we investigate several benchmark problems to demonstrate the efficacy of our proposed APNN methods.