I assume a nuclear thermal rocket with a solid core first. The principle of operation seems simple enough: Bring your propellant close to your nuclear fuel, start a reaction and wait for some heat/pressure to build up due to neutron (and other?) absorption. Once you have enough heat, you release the propellant through your nozzle and get acceleration, right?
However, when you release propellant, the pressure inside the reaction chamber drops, so you will have to refill it with some fresh propellant from your tanks. I consider this a severe practical problem, since the pressure inside the chamber is still quite high (ideally, if you want to achieve constant thrust, you need to work against your target pressure). From my limited understanding, it seems that you need to bring the propellant to an even higher pressure to press it into the reaction chamber, which would be absurd since then you could pump it through the nozzle directly.
So what am I missing here? How do you get the propellant into the reaction chamber? Bonus points for answering the same for the way more complicated liquid or gas core engines.