Well, for most of the Republic the consuls' primary duties were leading campaigns. There's a relatively recent book by Pina Polo, The Consul at Rome, that treats this topic pretty in depth, as you might guess from the title.

Pina Polo goes out of his way to point out that there's no consensus on the nature of consular imperium or potestas, and he's not really interested in that problem. Rather he focuses on consular duties at Rome, which seem to have been handled during the first week of a consul's term. Pina Polo doesn't seem to think anyone handled consular duties while they were out of town. For instance

Except in particular cases, they [the consuls] stayed in the Urbs only for the first weeks of their office, apart from the usual short stay of one consul to conduct the elections at the end of the consular year. Since it was the consuls - or at any rate, one of them - who had to present the prodigies to the senate and preside over the prescribed acts of expiation, it was necessary to wait until the new consuls took office in order to proceed.

If something absolutely had to be done, dictators could be appointed for specific purposes like dictator clavi figendi causa. This is extraordinary though.

After Sulla's reforms a lot of this changed. The use of more proconsuls and praetors meant the consuls spent more time in the city, some consuls not having any provincial appointments at all. Pina Polo thinks the evidence for regular consular practice in this period is a mess. Too many extraordinary appointments and such - apparently only Caesar is known to have conformed to the model.

If you want more info, there is also an edited volume, Consuls and Res Publica, that came out the same year as Pina Polo's book (and to which he contributes) that gets a little further into the debated issues.