Here is the passage in larger context:
ἀλλὰ καὶ οὔτε πόλιν οὔτε πολιτείαν, οὐ τέχνας, οὐκ ἐπιστήμας ἐπὶ νοῦν ἐβάλλοντο, νόμων τε καὶ δικαιωμάτων καὶ προσέτι ἀρετῆς καὶ φιλοσοφίας οὐδὲ ὀνόματος μετεῖχον, νομάδες δὲ ἐπ̓ ἐρημίας οἷά τινες ἄγριοι καὶ ἀπηνεῖς διῆγον, τοὺς μὲν ἐκ φύσεως προσήκοντας λογισμοὺς τά τε λογικὰ καὶ ἥμερα τῆς ἀνθρώπων ψυχῆς σπέρματα αὐτοπροαιρέτου κακίας ὑπερβολῇ διαφθείροντες, ἀνοσιουργίαις δὲ πάσαις ὅλους σφᾶς ἐκδεδωκότες, ὡς τοτὲ μὲν ἀλληλοφθορεῖν, τοτὲ δὲ ἀλληλοκτονεῖν, ἄλλοτε δὲ ἀνθρωποβορεῖν, θεομαχίας τε καὶ τὰς παρὰ τοῖς πᾶσιν βοωμένας γιγαντομαχίας ἐπιτολμᾶν, καὶ γῆν μὲν ἐπιτειχίζειν οὐρανῷ διανοεῖσθαι, μανίᾳ δὲ φρονήματος ἐκτόπου αὐτὸν τὸν ἐπὶ πᾶσιν πολεμεῖν παρασκευάζεσθαι: [20] ἐφ̓ οἷς τοῦτον ἑαυτοῖς εἰσάγουσι τὸν τρόπον κατακλυσμοῖς αὐτοὺς καὶ πυρπολήσεσιν ὥσπερ ἀγρίαν ὕλην κατὰ πάσης τῆς γῆς κεχυμένην θεὸς ὁ πάντων ἔφορος μετῄει, λιμοῖς τε συνεχέσι καὶ λοιμοῖς πολέμοις τε αὖ καὶ κεραυνῶν βολαῖς ἄνωθεν αὐτοὺς ὑπετέμνετο, ὥσπερ τινὰ δεινὴν καὶ χαλεπωτάτην νόσον ψυχῶν πικροτέροις ἀνέχων τοῖς κολαστηρίοις.
I don’t have a complete answer, but I think you’re on the right track. It’s really the meaning of ἐφ̓ that’s in question here; οἷς is simply vague and abstract. ἐφ̓ οἷς must be either temporal–simply “when”, “at the time when”, “in the era when”–or causal–perhaps “inasmuch as”. You might capture both the temporal and causal senses by translating “in the circumstances in which”. There’s no reason why ἐφ̓ οἷς couldn’t be both, and probably Eusebius didn’t give much thought to the precise meaning he intended to convey. I wouldn’t worry too much about it.
ἐφ̓ οἷς is not really transitional here, i.e., it’s not a relative expression used as a transition from the previous clause: οἷς has to be a true relative because the clause ἐφ̓ οἷς τοῦτον ἑαυτοῖς εἰσάγουσι τὸν τρόπον is subordinate to κατακλυσμοῖς αὐτοὺς καὶ πυρπολήσεσιν ὥσπερ ἀγρίαν ὕλην κατὰ πάσης τῆς γῆς κεχυμένην θεὸς ὁ πάντων ἔφορος μετῄει. If ἐφ̓ οἷς were transitional, it would introduce a clause that would in theory be subordinate to the preceding clause.
You might try looking at the various senses of επι in LSJ. None of these quite fits precisely, but you can see how this could be a loose adverbial relative expression of time or circumstances or cause, or all of those.
http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3De)pi%2F2
"in the circumstances in which they adopted this way of life, God, who oversees everything, was pursuing/would pursue them, like a wild forest scattered over/covering the whole earth, with floods and fires,. . . "
Note to self: μετερχομαι/μετειμι + acc. – “pursue”
The switch from historical present εἰσάγουσι to imperfect μετῄει seems somewhat jarring, to me at least.
Interesting importation of pagan mythology into Christian doctrine . . .