First of all a RG flow explicitly breaks conformal invariance. The point is what happens in fixed points. This is the result Zamolodchikov found and indeed $c_{IR} < c_{UV}$. At the fixed point this number $c$ is indeed the central charge of the CFT. Now, more generally an advance to the subject mainly in 4d QFT was made by Intriligator and Wecht using the $a$-maximization where they "almost" proved the corresponding theorem to Zamolodchikov's $c$-theorem of 2d (almost because of some subtleties of their technique, check their paper). In fixed points of the RG in 4d you get a CFT which is characterized by two central charges $a$ and $c$. In all worked out examples these guys found that indeed $a_{IR} < a_{UV}$ but $c$ was not behaving like this. Later it was found by Martelli, Sparks and Yau that the holographic dual of $a$-maximization, a procedure known as $Z$-minimization of cycles in tori, gave the same resuls, thus giving further evidence on the validity of the $a$-maximization results on $a_{IR}<a_{UV}$. In 2011 Komargodski and Schwimmer proved the $a$-theorem in its generality. Now, people are working a lot on 3d QFT to determine the above there as well using something called $F$-maximization, proposed by Jeffereis ( this is a nice talk by Niarchos). Now various $c$-extremization techniques are applied to various 2d CFTs as well like the (0,2) theory that arises from the low-energy dynamics of D3-branes wrapped on Riemann surfaces and M5-branes wrapped on four-manifolds. Therefore, the status of the $c$-theorems is subject to change daily. As you see most work is done by generalizations of the original idea of Intriligator and Wecth who extremize a parameter that coincides with the central charge of the CFT. From this one can draw various conclusions but the only proofs we have till now, at least that I am aware of, are the ones of Zamolodchikov and Komargodski/Schwimmer.