OK it depends a lot on what exactly you mean. There is a known problem with people talking about "carbon" / "atmospheric carbon" / "Carbon Dioxide" and using the terms imprecisely - so there is often a confusion about which exact chemical species is being discussed.
Tonnes of C to CO2 is (as you correctly stated) a conversion based on molecular mass, but this doesn't really help us if we're talking about something like methane which contains carbon, but has a dramatically different effect in terms of global warming. It is even more complicated when you take into account the change in lifespan of these different species (methane degrades in the atmosphere much more quickly than CO2 for example)
The usual method to deal with this is to calculate Global Warming Potential (GWP). This compares the actual radiative forcing over time (the change in the earth's energy balance) to a fixed time horizon (usually 100 years) when compared with CO2. We can therefore compare the actual equivalent effects of releasing a number of different gasses in units of CO2 equivalent.
The details of how it's calculated and why the time horizons are where they are is in the IPCC AR5 WG1 CH8 technical supplement (8.SM.11.1 Equations for the Global Warming Potential)
Hope this helps