When traveling across several countries in South America or Asia, do Dollars/local currency or Euro/local currency have better exchange rate than local currency of country A / local currency of country B in the same region?
Speaking from my experience, it is almost always better to keep a base amount of money in US dollars and convert it at each country to local currency. (Or, alternatively Euros or British Pound - but only if you already have them, i.e., no point going from Euros -> US dollars -> some other currency).
The reason for this is that US dollars are far more in demand - and thus exchange rates are more competitive - as compared to other currencies. In fact, with some Asian currencies you might find that isn't even possible to exchange one Asian country's currency to another without going through an intermediate currency like the US dollar.
While the exchange rate obviously fluctuates, I'll take a simple example to prove my point. Say you have $100 and you want to convert to Thai baht.
And you convert the same dollar amount to Vietnamese dong.
Finally, take the equivalent Thai baht and convert to Vietnamese dong.
The difference in this example is minimal (~1300 dong, in favour of US dollar conversion) but this example is based on stock exchange rates, which is almost never the case in real-life. Like I said, there's far more demand / supply of currencies such as US dollars (Euros, and somewhat, Pounds) - and more competition between foreign exchange dealers - so you're almost always guaranteed to get a better price with those, rather than converting your entire currency to a local currency of country A, and then in other countries.
Besides this obvious advantage, another reason to not convert your money to a regional currency is that due to exchange rates, the US dollar (.../ Euro / Pound) is easier to carry in terms of number of actual currency notes for any monetary value.
The question doesn't really make that much sense, as exchange rates vary continuously, however if you are looking to decide whether to take dollars to each country then change them there for the local currency, or whether to change all your money for local currency in the first country and then change to the next local currency as you travel, then I think this is answerable:
My decision would be entirely round risk: do you want to be carrying large amounts of currency with you? I would typically not take much currency, perhaps some travellers cheques, but instead go to one of the major financial centres on arrival in a country and transfer money from my home account into local currency. Worth hunting around for the best exchange rate that day, but worth looking at how reputable the exchange seems.