Global vs US stock analysis

I currently cover just the US (S&P 500 stocks in a sector). My coverage is now going global, so I will also be covering European, Asian, emerging markets, etc. For those of you who work in equity research, any advice on the advantages and pitfalls of global vs just US analysis? Do valuation discrepancies on a global basis narrow? Or do they not, partly because most of the people looking at those companies are only looking them versus other names in the S&P or EAFE or EM index? Thanks