<p>The rapid development of quantum networks has created an urgent need to investigate quantum steering in networks and its role in quantum information protocols. By leveraging joint measurability theory and the concept of steering-equivalent-observable (SEO) measurement assemblages, this work establishes two necessary and sufficient conditions (distinct from conventional inequality-based criteria) for determining the steerability of network states in an entanglement-swapping network. Additionally, since quantum steering operates via one-way local operations and classical communication (LOCC), such as local filtering on the trusted party, we establish a necessary and sufficient condition for the convertibility between steering assemblages under local filtering operations in the entanglement-swapping network. To enhance the practical relevance and readability of our theoretical results, we provide a concrete example in which incompatible measurements acting on the untrusted party are shown to necessarily induce network steering.</p>