How do institutional investors trade foreign stocks?

I work for a consulting firm primarily focusing on buy side business.

I have been working on an assignment, need some inputs from you all.

Question is how do institutional investors (read mutual funds, insurance, hedge funds etc) trade foreign stocks?

I know many houses have trading offices overseas and some use global equity trading services provided by brokers and some specialized execution management firms.

But how do they track execution? During US time when world market time zones are different and some Far East markets are closed.

Heard of some OMS and EMS platforms that allow routing trade order to global trading venues. But how do you keep track of all execution above all how do they assure they are getting best execution with active equities trading during respective market making hours internationally.

I know there are lot of aspects to it.. Any help clearing this up would definitely help.

Thanks.