How much current can a 0.1" pitch pin header pass?
Samtecs TSW standard 0.1" header has a variable voltage and current rating based on the matching connector it plugs into. This is due to how much surface area makes contact with the header. That says, it averages 4 to 6 amps per pin.

http://suddendocs.samtec.com/catalog_english/tsw_th.pdf

Based on the common design and material, most headers will have similar properties. 

When used for stacking, as in two boards soldered to the same header, the max amperage is 3 amps @ 80 degrees C (I'm assuming per pin). Except for some of their dedicated power strips which have 8+ amp.