The support of free trade is rooted in the concept that some nations produce some things better than others. If that's the case then, like for every other product produced within nation, we should specialize in what we produce best and other nations should specialize in what they produce, and then we will all be better off.
HOWEVER, there are some caveats:
-To some extent, the ability to produce things better is not natural, but artificially created by governments. These are distortions, and distortions CAN wreak havoc on the economy of a nation and all nations they trade with. China's low currency valued being maintained by massive holdings of Euros and dollars is one such distortion.
-"Free Trade" does not exist. Wealthier nations use their power to constrict the options of poorer nations. They have few options when a Western corporation dumps its goods on their markets, which causes a lot of harm to domestic producers, for example. The WTO, IMO, mitigates this to some extent, and poor nations can still gain a lot from free trade agreements.
-What about non-tariff barriers? Sometimes it can make good sense to subsidize an industry, but that's technically an unfair advantage given to domestic competitors.
-We can accomplish strategic objectives by selective "strategic" trade, IE, not selling advanced weapons to North Korea and Iran.
-Western nations still have large masses of moderately educated individuals (high school and below). Offshoring all their jobs and not worrying about the consequences may not viable, as these people may not have the ability to readjust back into the economy. That's not to say that we shouldn't allow offshoring of jobs; it means that we need to provide transitional help to affected people. And the problem may be a LOT worse in the coming years as technology improves and more jobs become offshorable.
-Not allowing foreign goods because they are unsafe or laced with diseases that may disrupt the ecosystem is a non-tariff barrier (IE, a barrier to free trade that isn't a tax), but it's still rational.
-The massive trade deficits that we have been running for a long time have led to creation of Soverign Wealth Funds. These are investment funds that are owned and operated by foreign nations. If they are smart investors, there's nothing to worry about, but if they start buying up all our defense assets or they are delibatrely starting financial panics, we may end up regretting not being protectionist.