Upgrading Cuirs to Cavs rarely is worth it. That does not mean that getting Rifling to get Cavs is a bad decision, IMHO some (newly built) Cavs to take out top defenders along with leftover Cuirs is the way to go, although I would in most cases prefer 2 Cuirs over 1 Cav, meaning that again troops>econ. Just Tech to Rifling without slowing down on building Cuirs. After that upgrading Cuirs to Cavs competes against further research and provides little gain, so I only do it if I plan to win on the tech level that has been reached. Also obviously a cav is better than a cuir.
Going to War against someone with Rifles with Cavs is possible, you just need enough troops. Taking out Rifles in the field is not much trouble, 50+% odds are obtained with 5xp units, promoted to pinch. The only problem is defenders in a city. Depending on the city (hills, culture) and the Rifles promotions I estimate losses at about 1 (40% culture, flat, CG1 Rifles) to 3 (80% culture, hills, CG3 Rifles) Cavs per Rifle. At this point in the game I estimate about 3-5 defenders/city + whatever the AI can build in reinforcements. Basically any such attack will incure (sometimes huge) losses, that does not mean that it is impossible. With your current empire that should be doable. In this game Charly who has cultural defences of 60-80% and a few cities on hills, with protective Rifles, is close to the worst scenario, this is the context in which I made my above statement.
[...] and then falling back in tech.
At this point this is not really a problem. You have three times as many cities as the AI, with Communism Research is not a problem. Also any AI can be taken out with Tanks and fighters/bombers (even Future era AI with all units unlocked), thus research for such war games is only mandatory up to industialism and flight, even radio is optional.
Really if you have three times the size of the AI and about tech parity you can choose your victory condition freely, except Culture, that would be slow, and Diplo, which does only depend on diplo (still possible in this game). So basically you can choose dom/conquest as war victories or space as builder/econ victory, the AI will not be able to stop you.