Headword and gloss are in the present indicative, third person plural. For the headword verb, e)gkaqi/zw, see generally LSJ s.v., illustrating its military and other senses.
[1] Despite Adler's marginal indication ('Thdr.'), there is no necessary connection with Theodoret (epsilon 65, epsilon 67). Rather, what is being glossed here might be Philochorus FGrH 328 F108 (the only extant instance of e)gkaqi/zousin as opposed to e)gkaqi/zousi), from the scholia to Euripides, Hippolytus 35: the Pallantidai do this at Gargettos (but Theseus eludes them). More probably though, the verb is extracted, and modified in this minor respect, from the quotation given: see next note.
[2] Quotation not identified by Adler, but it is Dionysius of Halicarnassus, Roman Antiquities 9.20.2 (probably via a lost part of the Excerpta Constantini Porphyrogeniti); again in part at epsilon 2358.