In isolation, any third-tone character should indeed properly be pronounced falling, then rising. In the five-level scale that is common in specifying tones for tonal languages, the typical characterization of the third tone is 2-1-4; that is, the end of the tone contour is higher than the start (though not as high as the first tone or the end of the second tone).
In conjunction with other characters in an utterance, however (as opposed to, say, a list of third-tone characters), the contour is truncated and it is, more or less, simply 2-1. Being the only tone with a "bent" shape to it, as it were, it takes longer to say, and the natural rhythm of the speech simply doesn't permit enough time to draw it out to its completion. I suppose in principle, it would be possible to speed it up so that the entire 2-1-4 contour could fit into the same "space" as a first, second, or fourth-tone character, but in practice, that doesn't happen. (For that matter, it may be that a character undergoing tone sandhi was originally just the "other half" of the tone contour—the 1-4 half, that is. It should be noted that the second tone is usually specified as 3-5, or sometimes 2-5.)
See pp. 45-46 in Robert Ramsay's The Languages of China and pp. 147-48 in Jerry Norman's Chinese for more details. (I'm afraid I'm an old fuddy-duddy who reads books as well as on-line materials, and this is what I have. Maybe you can find excerpts in Google Books.)
Norman's book, incidentally, includes a second case of tone sandhi that is not often described. He writes, in the closing of the aforementioned passage:
In addition to the cases of tone sandhi mentioned in conjunction with
the third tone above, one other minor case should be mentioned. In a
three-syllable combination, if the second syllable is a second tone
(either an original second tone or a raised third) and the first
syllable is either a first or second tone, the middle syllable changes
to a first tone in rapid speech...: [聯合國] liánhéguó "United Nations"
is pronounced liánhēguó, and [他買馬] tā mǎi mǎ "he buys a horse" becomes
tā māi mǎ. [The characters in brackets are my addition.]
I don't know if this is universal amongst Mandarin speakers, but when I read this, I was mildly surprised to find that I actually observe it in my own speech. It might be interesting to see how broadly this rule is observed.