<p>In this paper, we propose a new concept of convergence that unifies and extends both order convergence and unbounded order convergence, and we investigate its relationships with them. Many properties of (unbounded) order convergence are preserved within this framework. In particular, we present an alternative proof showing that unbounded order completeness implies universal completeness. Additionally, we show that this type of convergence is locally solid additive (and, in certain cases, linear), thereby preserving several properties of locally solid convergence structures that have been investigated recently.</p>