Is it more economical to liquify natural gas to ship overseas or convert it to propane (C3H8) and ship it overseas?
Gas to liquid technology is known, but propane isn't a good choice of end product

The idea of converting natural gas (which is mostly methane) into liquids is widely used not least because the earth has, inconveniently, located the best natural gas fields some way from the places that most want to use natural gas.

Right now (in 2016), though, it seems it is still not desperately uneconomical to convert the methane into a liquid by refrigeration and ship it to customers in the largest known thermos flasks (LNG tankers). Both the ships for transporting liquefied natural gas (LNG) and the plants to create it are very expensive (tens of billions of dollars for the minimal economic scale). There are something like 410 of these ships and there seem to be dozens of new units being built every year (see this report on the industry for more detail: warning pdf).

But there are some examples of plants that convert the local gas supply directly to more transposable liquids. Shell, for example, have developed a process that converts gas directly into clean liquid fuels (the initial plants produce very clean diesel, but other fuels are possible). See this description of the Shell process. Some other firms are operating similar processes.

So it seems that, right now, LNG is the best economical option. But if you are going to use a technology to go directly to something more transportable than methane, it is better to go the whole hog to a widely used fuel (diesel and petrol are both basically longer chain hydrocarbons than propane).

This may, of course, change.