This paper shows the convergence of adele-valued random walks to an adelic Lévy process under scaling limits. We use random walks on the $p$-adic numbers to construct random walks initially on the infinite product space, and use survival time analysis to prove that the random walks are almost surely adelic for all time. The adelic random walks are shown to be small perturbations of processes that are supported on a finite product of path spaces. Weak convergence to an adelic Lévy process is established in the $J_1$ Skorokhod topology.