In this paper, the authors give criteria for relatively compact sets in Banach space-valued bounded-variation spaces in the sense of Jordan, and Banach space-valued bounded Wiener $p$-variation spaces when $p \in (0, 1)$. Then, they give sufficient conditions for relatively compact sets in other Banach space-valued bounded-variation spaces, such as Banach space-valued bounded Wiener $p$-variation spaces when $p \in (0, 1)$, Banach space-valued bounded Wiener-Young variation spaces, Banach space-valued bounded Schramm variation spaces, Banach space-valued bounded Waterman variation spaces, Banach space-valued bounded Riesz variation spaces, and Banach space-valued bounded Korenblum variation spaces. \par In the paper, the definitions of the spaces mentioned above are given one by one. In each section, the results have the following form: \par Theorem: A set $A \subseteq {\cal S}(I, E)$ is relatively compact if and only if (or if) the following conditions are satisfied: \par (i) $A$ is bounded; \par (ii) the set $A - A = \{x - y \mid x, y \in A \}$ is equivariated; \par (iii) for every $t \in I$, the set $\{x(t) \mid x \in A\}$ is relatively compact in $E$. \par Here $I = (0, 1)$, $E$ is a Banach space and $\cal S$ is each of the above-mentioned spaces. \par It should be noted that only conditions (i) and (ii) are strictly related to the space ${\cal S}(I, E)$.