Locally in/out-semicomplete digraphs are important generalizations of semicomplete digraphs.By using the theory that every locally in-semicomplete digraph is in-path-mergeable

and employing the multi-insertion approach for digraphs

the authors have obtained the sufficient conditions for a locally in-semicomplete digraph to be traceable under degree-constrained condition.The authors have also obtained a sufficient condition for a locally out-semicomplete digraph to be traceable by taking advantage of the property that the converses of locally out-semicomplete digraphs are locally in-semicomplete digraphs.