In this paper we study abstract elementary classes with Löwenheim-Skolem number $\kappa$, where $\kappa$ is cofinal with $\omega$, which have finite character. We generalize results obtained by Kueker for $\kappa =\omega$. In particular, we show that $\mathbb{K}$ is closed under $L_{\mathrm{\infty },\kappa }$-elementary equivalence and obtain sufficient conditions for $\mathbb{K}$ to be $L_{\mathrm{\infty },\kappa }$-axiomatizable. In addition, we provide an example to illustrate that if $\kappa$ is uncountable regular then $\mathbb{K}$ is not closed under $L_{\mathrm{\infty },\kappa }$-elementary equivalence.