A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly distributed on $[-1,1]$, then the random variables $f(L\cdot V)$ converge in distribution, as $L\to\infty$, to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes $(f(L\cdot V+t))_{t\in\mathbb{R}}$ in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. We further investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of [Limiting distributions of the classical error terms of prime number theory, Quart. J. Math. 65 (2014), 743--780] and earlier works.