If you have cut the enemy's links to metals, then sometimes you have to engage in a bit of a longer war as opposed to slamming into a rather highly defensive city.
Usually you can surprise an AI and take a city or two initially which takes a bite into their overall production. However, they will usually continue mass producing units usually concentrating them in the most threatened cities.
Nice thing about mounted is you can quickly move around the territory, using tactics to better your situation. If a city..like the cap..appears to be heavily defended such that an attack might prove devastating at the moment, instead try drawing out defenders to attack them in the field. Use forking or simply move toward another city to draw out defenders until you wear them down enough to successfully take the city. Another option is using a ceasefire (not peace treaty), which may relax the enemy, who will then start sending out workers and units from within the cities. Attack again in a turn or two hitting those units immediately.
(just to clarify my position on barracks: yes, I usually, if possible, try to get up barracks at least in most cities I can. However, usually there may be 1 or 2 cities, depending on how many i have that may be better in just producing HAs immediately at that point. Stables on the other hand are a bit of an investment and you can/should forego building them unless you have time. In other words, if HAs are now online don't wait to build stables to start producing them.as well as barracks really at that point)