Authors’ abstract: Given a uniform space

$X$ and nonempty subsets

$A$ and

$B$ of

$X$, we introduce the concepts of some families

$\mathcal{V}$ of generalized pseudodistances on

$X$, of set-valued dynamic systems of relatively quasi-asymptotic contractions

$T:A\cup B\to {2}^{A\cup B}$ with respect to

$\mathcal{V}$ and best proximity points for

$T$ in

$A\cup B$, and describe the methods to establish the conditions guaranteeing the existence of best proximity points for

$T$ when

$T$ is cyclic (i.e.,

$T:A\to {2}^{B}$ and

$T:B\to {2}^{A}$) or when

$T$ is noncyclic (i.e.,

$T:A\to {2}^{A}$ and

$T:B\to {2}^{B}$). Moreover, we establish conditions guaranteeing that for each starting point each generalized sequence of iterations (in particular, each dynamic process) converges and the limit is a best proximity point for

$T$ in

$A\cup B$. These best proximity points for

$T$ are determined by unique endpoints in

$A\cup B$ for a map

${T}^{\left[2\right]}$ when

$T$ is cyclic and for a map

$T$ when

$T$ is noncyclic. The results and the methods are new for set-valued and single-valued dynamic systems in uniform, locally convex, metric and Banach spaces. Various examples illustrating the ideas of our definitions and results, and fundamental differences between our results and the well-known ones are given.