The problem with this topic is that it is just too large to handle without writing a book.
The closer you can get to what you describe is through ADRs or the reciprocate systems. Such systems are allowed on most large exchanges.
These systems allows the listing of a title which is actually backed by the original security.
A couple of examples :
Alcatel(CGE:Euronext) has an ADR(NYSE:ALA)
STMicro(STM:Euronext) has an ADR(NYSE:STM)
Suez(SZE:Euronext) has an ADR(NYSE:SZE)
Vivendi(EX:Euronext) has an ADR (NYSE:V)
France Telecom(FTE:Euronext) has an ADR (NYSE:FTE)
IBM (NYSE:IBM) has an Euronext equivalent (IBM:Euronext)
AT&T (NYSE:T) has an Euronext equivalent (ATRP:Euronext)
Altria (NYSE:MO) has an Euronext equivalent (PHM:Euronext)
Same thing for a whole bunch of companies...
Now, the actual processing of this kind of trading requires a good understanding of the trading rules of both exchanges you have to work with. (for example, the leverage rules, odd lot rules and short rules are completely different between Euronext and the NYSE). Then, other factors such as cost of carry, cost of credit, income on debit and risk relating to holding a part of the account in a different currency have to be factored in.
Sometimes, it is better to go directly, sometimes better to build permanent long and short inventories.
Except(maybe) if your broker can clear for all the exchanges you trade, you will have to negociate (at a cost).
Taxes are also something to think about very seriously, probably the most important thing to check.
Well, maybe this will give you a couple ideas for the start.
Your humble servant,
OHLC