It seems to me that there is really no puzzle here, just many fuzzy definitions for trade and investment flows between countries. If goods are produced within the borders of a country by a foreign-owned firm and then exported out of the country, this flow is recorded in the balance-of-payments of the foreign-owned firms's country as an export. Thus, a country can have increased exports without increased production (no increased domestic employment) within the borders of the country. Of course there are local content requirements that can make this all sort of fuzzy.
This is one outcome of foreign direct investment. FDI has actually increased faster in some instances than trade itself, so global manufacturing and employment inconsistencies are really no surprise. One indication of this is to look at the difference between GDP (location valued production) and GNP (ownership valued production) in a country like China. There is a great deal of production within the borders of China but who OWNS the factors (capital) that produce it?
Diane