To discuss the prosody of spoken English one often has to resort to the language of music: pitch/intonation, rhythm, melody, phrasing, volume/dynamic, and in this instance, cadence.
Just as in music, in rhetoric there are different kinds of cadences: a perfect or full cadence, in most speakers with a falling pitch to signal the end of a sentence — cadence derives from the Latin verb cadere, ‘to fall’ — and an imperfect or half cadence, usually with a slightly rising pitch, though different from that when asking a question.
In general, the extra vowel, most often a schwa and always with falling pitch, does not randomly occur at the end of just any word, but emphasizes the end of a phrase/clause, sentence, or paragraph — in essence, an audible comma or period to mark a cadence.
One could also think of it as the vocalization of a normally silent or barely audible consonantal release or in the case of stops, aspiration. To my ears, to pronounce the extra vowel in a vowel-final word, the air flow must be stopped briefly to articulate it. As such, it can be found in a wide variety of oral speech.
In the fifth season finale of the new Doctor Who, Amelia Pond (Karen Gillan) cries out to the Doctor (Matt Smith): “Raggedy Man, I remember you, and you are late for my wedding-uh!” In this case, it marks the cadence of an entire speech which has increased in volume and intensity before the Tardis materializes.
I remember this instance especially because I do not generally associate this feature with British English, but rather a particular kind of revivalist/evangelical preaching in the American South:
The roving evangelist would have a sawdust oval laid out and a circus tent erected over it, with folding chairs and an elevated platform with pulpit, and before the platform a clear spot for the healees to throw crutches or walkers or dark glasses and canes, ear horns, once the evangelist had laid a hand on their foreheads and yelled, “Heal in the name of Jesus-uh!” — Paul Ruffin, “Harvey Watson and the Angel,” Jesus in the Mist: Stories, Columbia SC, 2007, 14.
After a slight warm-up an 11-year-old preacher often punctuates a lengthy hymnic/lyrical section with these extra vowels and always with an audible intake of breath, the prosody underlined by a musical accompaniment much in the manner of a recitativ.
As far as I know, this feature has no technical designation, but as it is most commonly heard marking cadences, cadence marker would do adequate service, or with more detail, a schwa-augmented cadence might be a good suggestion.