A linear order $(L, <)$ is $\lambda$-transitive iff any order-preserving bijection between sets of size $\lambda$ can be extended to an order automorphism of $L$.
For $\lambda < \aleph_0$, $2$-transitive orders can be found as the reduct of an ordered field and $2$-transitive implies $k$-transitive for all $k < \aleph_0$. I'm curious about what happens when $\lambda$ is infinite. In particular, do $\lambda$-transitive linear orders (or something like them) exist?
Some additional restriction seems necessary: different order types or map between a sequence with a lub and one without would cause problems. If the definition requires the sets to be inf and sup closed, does this help?