There is no inherent link between free trade and democracy specifically. Free trade has a tendency to improve economies, and that seems to lead to systems of government where the base of power lies with the people, but that does not mean the nations adopt democratic systems. However, there is a direct tie between free trade and democracy because "first world" nations are largely democratic. In order to gain access and status, it is in a poorer country's interest to adopt a system of government that pleases the richer countries.
The idea of free trade promotes a more global unity. Countries that support free trade see greater economical advances and also more advances in human rights. Countries that open up to free trade are more likely to be successful than those who continue to live within borders closed to free trade.
Free trade promotes democracy in the world because the more freedoms that individuals and companies have, the more democracy is going to flourish. Countries should promote open and free trade because while some protectionist restrictions are necessary, in general, the more regulations you have the more you are infringing on freedoms and not promoting democracy.
Despite what the term 'free trade' suggests, it is inherently undemocratic because organisations such as the International Monetary Fund and the World Trade Organisation lack accountability and enforce a globalist agenda on it's members whilst bullying non-members into forceful membership and/or economic reform.(See the Dissolution of Yugoslavia, where the IMF and the WTO saw Yugoslavia's market socialism as an affront to their globalist policies and bullying China to move from a fixed exchange rate system to a managed exchange system.
Free trade permits multinational corporations to increasingly exploit resources of underdeveloped countries in the name of 'liberty.' This couldn't be further from the truth. Liberty for the capitalists, yes, but none for the workers and the indigenous population. It does not lead to democracy but rather a global empire run by the oligarchy. This has even been confirmed by former economic hitmen who used to work for private businesses. They'd set up regimes that are friendly to the US and its 'good intention' of democracy and pump all the resources out of the third-world countries to fuel the first. Free trade just removes a burden on the oligarchs' back.