Note: I'm not an expert on this topic, but since no one has made a reply in 11 hours and I have some knowledge, I hope I can add something. This is just my interpretation of all I've read rather than being inspired on a few books analysing the military history of the continent. If that's unacceptable, I apologise in advance.

To begin with, wars in both continents were not similar nor could they be, for a variety of factors:

1- The wealth and population: Naturally, Europe was far superior in both aspects to South America to the point that the armies of entire countries, aside from often being mostly unprofessional militias or cavalrymen, numbered in thousands (compare with the massive battle of Borodino, with over a hundred thousand soldiers on both sides). The one exception to this rule was the overly centralized and militaristic state of Paraguay up to the end of the Paraguayan War, which could dispose of approximately 50.000 men for the conflict (amount that still pales in comparison with Europe). Now, this is not to say artillery or professional armies were non-existant, but their proportion was not that of a European army. Railways were also lacking.

2- The weakness of the State: Related with the previous point. South America in the 19th century was made up of numerous local governments struggling to find their place in the world, and not only their structure, but their very existence was contested from the inside. As a matter of fact, the vast majority of wars of the century were local, not international, and even in those cases the former were involved (look for example to the Platine War, where the trigger was long Uruguayan Civil War, itself related to the decades-long conflict within Argentina).

3- Geography: Due to its size and also to its features, in particular the Andes and the Amazon, but also other marshes; many regions are either remote or isolated, in sharp contrast with most of Europe. As a result, the wars of movement Napoleon favoured were usually out of the question, bar some instances. In a similar fashion, the quick seizure of "key points" like a capital became impossible unless they were close to the border. Finally, climate was another concern for any expedition (If I recall correctly, most losses in both San Martíns expedition through then Andes and the Allied invasion of Paraguay have been attributed to Geography rather than the battles themselves). Furthermore, the large areas at play make the scarce railroads even less useful, to the point that during the War of the Pacific (the last major conflict fought entirely in this century) Chilean forces invaded Perú mostly by means of amphibious attacks.

To address question itself (how were the wars fought) let me first say there were changes in this long period (as could be expected). With that said, I'll summarise the key points that haven't already been mentioned:

*Originally you have the colonial armies loyal to their governments meant to fight like those of the motherland *The collapse of Spanish power leads to caudillos who have their own unprofessional armies *In the last decades the independent states begin a process of stabilisation and professionalisation of their armed forces, consequences of this are the final defeat of the Caudillos and the Naval Arms race between Chile, Argentina and Brazil. *The first half of the centuries sees many Europeans fight in the Americas, often as leaders. Examples are San Martín, William Brown, Garibaldi and Cochrane.

I hope that is useful.