After the caravan arrived in the destination city and created a road, it should disappear(or rather, become that little trader which then moves back and forth on the road). If it did not, its a bug. The little trader has two effects: It provides a % bonus to food production in both connected cities(so it makes sense for at least one of them being a food-producing center), and it levels up the road(i'm not sure what the difference between level 1 and 2 roads is tho). Its all automatic, you dont need to do anything further save make sure the little guy does not get ambushed by monsters on the road.

Trade is possible through the diplomacy screen, talk to another faction, offer them your excess goods in exchange for something else, cash or otherwise.

1) that is all that you do with the caravan. Once it reaches the other of your cities the road is created, the "little trader" begins going back and forth and the result is a bonus to food production. Trade route is established. unless/until the caraven is attacked and destroyed by enemies or random monsters

2) you cant move it to another city once it establishes a route and you can only control one caravan per city till you research "advanced trade". That increases the number of caravans each city can build and send out.

3) There is no need to trade between your own cities because food, metal, materials, and later on crystals, horses, etc  are all "global" resources and shared by all of the cities in your kingdom/empire.  So if you have a city that produces a net 5 food per turn and your second city (possibly with no food source) consumes 2 food per turn, as a kingdon you have a net 3 food per turn surplus.

4) the only way to turn these into cash is to trade them with another faction and ask for gildar in return.

Hope that helps!!!

Saying there's no point trading between your own cities is incorrect. You benefit just the same from the increase in food from trade routes.

As for what the more advanced roads do, they increase the % in trade.

To the OP's original question, I have experienced what you describe, and I found that sending it to a building within your other city is not enough the teamster driving it appears to be a card-carrying-union-member who only does and goes exactly to where you send him and if you do not send him to the original settlement tile WITHIN your city then he'll just park and do nothing... hence my trouble with unions...But I digress.

I've mentioned this before but it hasn't been confirmed nor denied:

When I established a caravan between 2 cities (one with -3 food and one with +8 food) the food levels went to -4 and +9 respectively. I have theorized that the multiplier is unaware of the original number being negative and therefore food improvements actual increase the negative amount produced.

I've only played one game and hadn't been brave enough to raze one of my irrigation systems yet to see if this is accurate or if something else is afoot.

Anyhow, Just my 2 cents.