One could extend Kevin's examples endlessly. Most history is still local history - with the sense of the "local" taking all sorts of forms - a town, a region, a state, a nation, etc. Why the global aspect should take precedence, either conceptually or politically is beyond me.
Professor Fernlund is correct to warn against conflation of global history and "globalization," the latter word politically-loaded because it is rarely defined and therefore means who-knows-what? To my knowledge, there are only two arguments for the "birth of globalization" in the economic history literature. O'Rourke and Williamson have proposed a birthdate in the 1820s, based up an economics-only (Eurocentric) viewpoint narrowly limited to prices of less than a handful of commodities.