Sure they can adjust output. Some are not licensed to do so, but that's more a paperwork hindrance than any actual physical reason. They can ramp about as fast as other generators using a steam cycle, like ccgt or coal. There are some situations like when you're at the end of the fuel cycle if you ramp down quickly you might have trouble starting quickly again due to Xe poisoning. But, if there was a reason to, the control system can be designed to provide sufficient excess reactivity to power through events like that (e.g. naval reactors).

That being said, since most of the cost of a nuclear power plant is the capital cost, every time you're not generating at full power is time you're generating less revenue to cover the capital costs. So even in a hypothetical lowest cost zero carbon system with very high penetration of variable renewables (solar, wind), the nuclear plants would be mostly running flat out.