We consider analogies between the logically independent properties of strong going-between (SGB) and going-down (GD), as well as analogies between the universalizations of these properties. Transfer results are obtained for the (universally) SGB property relative to pullbacks and Nagata ring constructions. It is shown that if $A\subseteq B$ are domains such that $A$ is an LFD universally going-down domain and $B$ is algebraic over $A$, then the inclusion map $A[X_1,\cdots ,X_n]\hookrightarrow B[X_1,\cdots ,X_n]$ satisfies GB for each $n\ge 0$. However, for any nonzero ring...