We know that we trade dollars for the goods and services we desire from the other side of lines drawn on maps. A couple trillion a year of those dollars returns to purchase stuff---like wheat, rice, soybeans, poultry and dairy---produced on this side, the remainder returns as investment in assets. I.e., we get the stuff we want while creating business for U.S. exporters and opening markets for U.S. assets. Trade is a beautiful thing: it is the path to prosperity, and peace*. As consumers, and voters, breaking down protectionist trade barriers should be at the very top of our list of priorities.
So what would make trading natural gas any different? Unequivocally nothing!! There is no legitimate case, economic or otherwise, for restricting the sale of natural gas to other markets. Any attempts at such, as convincing as they may seem ("higher demand means a higher price"), are led by those (like America's Energy Alliance, a group formed by Dow Chemical, Alcoa and Nucor) whose execs believe, erroneously, that their companies benefit by restricting global opportunities for U.S. energy producers (I wonder how loud they'd scream if new export limits were placed on them). They don't seem to understand basic economics: that restricting the export of natural gas can only result in the industry, which is counting on the opening of trade routes, restricting production---which results in what? You got it; a higher price. That is, a higher price without all the economic growth (think jobs) that would result from free trade. Although I suspect that they clearly understand (and would trumpet if need be) that most basic concept when applied to the export of their respective commodities.
*"Greatly increased levels of international trade and foreign direct investment have raised the costs of conquest and shrunk its benefits. In today's open global trading system, it is almost always cheaper to acquire goods and raw materials by trade than to invade a country in order to steal them." The Human Security Report
*"When goods do not cross borders, soldiers will." Frederic Bastiat