We study the long-time dynamics of the Navier-Stokes equations in the threedimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey-Sobolev spaces in such a general system, then any Leray-Hopf weak solution admits an asymptotic expansion of the same type. This expansion is uniquely determined by the force, and independent of the solutions. Various applications of the abstract results are provided which particularly include the previously obtained expansions for the solutions in case of power decay, as well as the new expansions in case of the logarithmic and iterated logarithmic decay.g(x + Le j ) = g(x) for all x ∈ R 3 , j = 1, 2, 3, where {e 1 , e 2 , e 3 } is the standard basis of R 3 , and is said to have zero average over the domain Ω = (−L/2, L/2) 3 if Ω g(x)dx = 0.By using a particular Galilean transformation, see details in, e.g., [19], we can also assume, for all t ≥ 0, that f(x, t) and u(x, t), have zero averages over the domain Ω. In light of the Leray-Helmholtz decomposition, and for the sake of convenience, we also assume that f(x, t) is divergence-free for all t ≥ 0.By rescaling the variables x and t, we assume throughout, without loss of generality, that L = 2π and ν = 1.Throughout the paper, we use the following notation u(t) = u(·, t), f (t) = f(·, t), u 0 = u 0 (·), which are function-valued.