Travel from New Orleans to Dominican Republic via Miami

The US does not have much in the way of outgoing immigration controls. They've "convinced" the airlines to collect and pass the information to them electronically, so non-citizens who used to have an I-94 or I-94W form no longer receive a form that they would have had to drop at the last US airport (Miami in your case).

The airline should check your documents to make sure it looks like you're not likely to be rejected by DR customs- passport etc. (if you were a citizen of, say, Hong Kong, they'd check for the presence of a visa). I think they'll do that in New Orleans, but if you booked the flights separately on two airlines, maybe not.

http://www.cbp.gov/xp/cgov/travel/id_visa/i-94_instructions/i94_rollout.xml

On the trip out, immigration will take place when you arrive in the Dominican Republic. There is no specific procedure when exiting the United States, you only need the usual time to change planes (including transferring between terminals).

On the way back, you will go through US immigration when you land in Miami. There are no international transfers in the US, everybody goes through immigration as soon as they land. Allow a generous buffer, especially if you aren't a US citizen.