War. War never changes has multiple causes, some more evident than others.
The Paraguayan War (down South, we usually call it 'War of the Triple Alliance', or if you are feeling revisionist, 'War of the Triple Infamy') has a long, storied debate on its causes - in particular the British involvement, or lack thereof. Classical historians usually make Solano López' aggressive politics and attempts to meddle with Uruguayan politics the cause of the war, revisionist history places the blame on British interests and imperialism. The truth is, probably, somewhere in between.
Disclaimer: For convenience, I will use the current name of the countries involved. Instead of the several iterations of Provincias Unidas, Confederación Argentina I will talk about 'Argentina', instead of Banda Oriental or Cisplatina I will refer to Uruguay, and instead of 'Misiones Orientales' I'll go to 'Misiones'.
Now, for some intro context. After the Independence Wars, South America was unstable riddled with relatively small interstate wars, and widespread civil unrest. Conflict over Uruguay led to the Brazilian-Argentine war, and was settled on an independent Uruguay. Paraguay, a former royalist stronghold, remained as part of Argentina - while operating as an almost independent state. It only declared its independence on 1842.
The first issue to be considered is probably that the independent Paraguayan state inherited the poorly defined boundaries conflicts that plagued Spanish-Portuguese relations from the times of the Conquista. Even by 1850, there had been conflicts regarding the Tacurupyta yerba mate zone. A neutral zone was proposed as a solution to the border conflicts, but the treaty was quickly dismissed by the Empire of Brazil towards 1860, with attempts to found military colonies that sparked several border clashes. Once the war was finished, Brazilian and Argentinean claims on Paraguayan lands were pressed. This is reflected on the treaty that formalized the Triple Alliance, on Article XVI, to reflect the 'proper' boundaries.
ARTICLE XVI
The second issue, and one of the main contention point for liberal, classical historians, is the involvement of the two South American powers in Uruguay. That is, the support that López gave to the White-fusionist (federalist) party in the Uruguayan civil war, while the Argentine and Brazilian governments supported the Colorado (unitarian) side, as well as making overtures to former Federal strongman Justo José de Urquiza in Argentina, was considered meddling on traditional Brazilian and Argentinean spheres of influence.
The third cause, and to this humble amateur the most important, is that the protectionist, pro-industrialization internal measures of Paraguay - including tariffs and blockages on river navigation - were seen as a threat to the Argentinean and Brazilian interests as main continental powers. Article VIII of the treaty is clear on this:
The present Government of Paraguay being overthrown, the Allies will proceed to make the necessary arrangements with the authority constituted, to insure the free navigation of the Rivers Parana and Paraguay, in such manner that the regulations or laws of that Republic shall not obstruct, hinder nor burden the transit and direct navigation of the merchantmen and vessels-of-war of the allied States proceeding to their respective territory, or to territory not belonging to Paraguay, and they will take suitable guarantees for the effectiveness of those arrangements
We also have Mitre's words in this regard, on his 1869 speech to the traders in Buenos Aires:
In the War in Paraguay, that has ended or can be considered as finished (...) Argentina has triumphed (...) not only in vindication of rights, but also in the principles of free trade, which vivify commerce..
The fourth cause, considered the 'cutting down to size' of Paraguay and the prevention of a third continental power from rising, can be viewed in the first two Protocol clauses of the treaty. It's a disarmament clause, and it hankers back to the third clause. Humaitá was an island fortress that could be used to impede free navigation and trade.
- That in fulfilment of the Treaty of Alliance of this date, the fortifications of Humaitá shall be caused to be demolished, and it shall not be permitted that others of an equal nature should be erected, which might impede the faithful execution of that Treaty.
- That it being one of the measures necessary to guarantee peace with the Government that shall be established in Paraguay not to leave arms or elements of war, those that are met with shall be divided in equal shares between the allies.
As for British involvement, well. The main argument from revisionist schools is that the British were seeking cheap cotton after the defeat of the Confederacy - and that they pressured the Triple Alliance into going to war for this reason, to open Paraguayan markets. This is, generally, regarded as a myth. There is no documentation that supports British involvement in the war. Brazil broke relations with the UK five months after the war began. Paraguay was already a big customer of British companies for their modest effort at industrialization. While doubtlessly the British profiteered of the war, as any world power is wont to do, and supported the Triple Alliance politically, there is little that suggests direct or indirect involvement aside from the usual meddling in their spheres of influence.
I hope I was able to be brief and clear. The Paraguayan war was a complex, multi-causal conflict, that brought a lot of pain and misery to South America, and remains one of our most controversial historical topics. Let me know if I can expand or clarify in any way anything!
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