  Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear
order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it
is possible to partition $L$ into convex sets indexed by some element of
$\mathcal{L}$ which are isomorphic to convex subsets of $L'$ ordered in the
same way. This notion generalizes convex embeddability and (finite) piecewise
convex embeddability (both studied in arXiv:2309.09910), which are the special
cases $\mathcal{L} = \{\mathbf{1}\}$ and $\mathcal{L} = \mathsf{Fin}$. We focus
mainly on the behavior of these relations on the set of countable linear
orders, first characterizing when they are transitive, and hence a quasi-order.
We then study these quasi-orders from a combinatorial point of view, and
analyze their complexity with respect to Borel reducibility. Finally, we extend
our analysis to uncountable linear orders.
