Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$).
Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a maximal set of points which in finite distance to one another).
We say that $d_1$ is asymptotically bounded by $d_2$, and denote it by $d_1\preceq d_2$, if for any $A\subseteq X^2$, if $d_2$ is bounded on $A$, then so is $d_1$.
Maybe more generally, $d_1$ is componentwise bounded by $d_2$, denote it by $d_1\preceq' d_2$ if for any set $\mathcal A$ of metric components, if elements of $\mathcal A$ have uniformly bounded (below $\infty$) diameters with respect to $d_2$, then they have uniformly bounded diameters with respect to $d_1$.
It's easy to see that $\preceq,\preceq'$ are partial orders and ${\preceq}\subseteq {\preceq'}$ (though in general the inclusion is strict – ${\preceq'}$ is total if all components have infinite diameter). It's also easy to see that both of these partial order has a minimal class: that of a metric which is just discrete $0$-$1$ metric on each component.
Does either of $\preceq,\preceq'$ (or maybe some closely related concept) have a well-established name? I'm interested in some references about this kind of comparison, in particular about the existence of infima.