Assume $\kappa$ is uncountable and $\phi$ is an $L\_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec\_{\infty,\kappa}$. It is well-known that $K$ is closed under unions of increasing chains of length $\kappa^+$, but the argument fails for increasing chains of length $<\kappa$. 


Do you have an easy counterexample, either for $\kappa=\aleph\_1$ or $\kappa=\aleph\_\omega$?



Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains
In $L\_{\infty,\kappa}$, you can express that a linear order has cofinality at least $\kappa$, since you can say that every less-than-size-$\kappa$ family is bounded. But having cofinality at least $\kappa$ is not generally preserved under unions of elementary chains of length $\lt\kappa$, although it would be for longer chains. 


