Why do you "carry out" an action in English? It's not really related to the base meaning of "carry". In German, the corresponding verb is "ausführen" ("führen" = to lead, so the image seems to be that you somehow "lead" the action to where it is supposed to be). The Latin word "executare" (to execute) contains "ex-sequi", i.e. "to follow out"; the reverse image. So even in western languages, there's no simple relationship between the base meaning of the words used in this particular expression. http://www.kanjinetworks.com (you need a free account to look stuff up) explains the meaning of 行 as follows: go; movement; procession; road → line (of people) → conduct; do; perform (← carry out an action)

行うis equal to おこなう｡行くis equal to いく｡行くmeans go.... 行うmaybe means hold.