I've seen the claim that when it comes to tonal contours, contour tonal languages "select from as many categories as possible" (Routledge Linguistics Encyclopedia). They posit the four basic tonal shapes as: level, rising, falling and convex/concave; naturally, standard Mandarin's four tones fit into one of each of these.
However, this is noticeably absent from standard descriptions of any Yue Chinese group (which includes Cantonese and Toisanese as well as the 10-tone Bobai dialect) that I'm aware of, as well as from the main Hakka varieties and from the Hokkien branch of the Minnan Chinese group. Both of these groups have more than four phonemic tones (although the Teochew branch of Minnan has a low falling-rising tone for the 阴去 category).
Many Mandarin varieties, not just standard 普通话 / 國語, have a concave tone. From Wikipedia's page on the four tones of Middle Chinese, Jinan in Shandong, Chengdu in Sichuan, and Nanjing in Jiangsu all share this feature with Beijing. The Nanchang variety of Gan and the Wenzhounese variety of Wu seem to also have concave tones (Wenzhounese having this longest of contours reliably paradoxically in the historical 入声 tones, which are the shortest tones in most of the rest of south China!).
Of those rarities that have both, Fuzhounese of the Min Dong family is probably the most well-known Chinese variety, with the 阴去 tone a concave one, whilst the 阳去 tone is convex. However, to my ear the concave tone is subject to a similar "smoothing" as in standard Mandarin, whilst the convex tone is somewhat more salient. The status of the tones in Suzhounese of the Wu family is subject to debate (as with all the tones of Wu really - tone sandhi has just become too complicated there!): the concave one is well-attested, but the convex one is allophonic to... something else (either low rising or low falling).
Leaving behind the Sino-Tibetan for the Oto-Manguean, Yoloxóchitl Mixtec has both too, although I'm not sure what other tones there are.
Just glancing at the literature for Vietnamese phonetics, the case with Vietnamese phonation and pitch-register is particularly interesting, as the "citation" concave tone "hỏi" is probably the one "tone" where the pitch contour is most important; the breathiness has been analysed has merely a phonetic realisation. However, it seems that it may be confused with huyền in Hanoi where the rising end of the hỏi contour is weakening. Southern Vietnamese on the other hand does not have a canonical concave tone: hỏi has merged with ngã to form a low-rising tone.
What generalisations can be drawn then? Common ancestry? Areal effects? Even support for the maximisation of the distribution of tone shapes is shaky. The only commonality would be that rising and falling tones always exist alongside the concave and convex.