Also, to add to the other poster's point, in a medieval siege, the defenders have every reason to believe the attackers will happily let every man, woman, and child behind the walls die gruesome deaths to starvatiom or disease. That's why, when it came down to the wire, cities would submit.

In modern times, cultivating a believable military posture of, "Surrender, or we will personally execute every last motherfucking one of you" is politically dicey. Look at the news stories coming out of Gaza about supplies running low thanks to Israeli interference. Right, wrong , or indifferent, the international community (as well as your domestic community, if those that disagree with these sorts of tactics are allowed to make their voices heard) tends to look down their noses at targeting noncombatants populations. So, due to these complications (which were largely absent or less impactful from warfare in the time of Genghis Khan) wholesale slaughter of civilian life isn't really openly used. In fact, guidelines like "proportionality" are invented which dictate the level of response you can give certain provocations and what not.

So, if you're a modern day commander being tasked with taking an urban center, the closest way to approximate a medieval siege would be to absolutely carpet bomb everything. Make it known that you will happily let every single person in Moscow die, if not send them to the afterlife yourself. While you're bombing the suburbs, you'll also need to encirce the whole city to prevent supplies from being delivered, since you can't guarantee every bomb will hit it's target and need starvation to provide additional assurance to the population that, if they maintain their current course, they are doomed.

Unfortunately, the world isn't going to allow that, and you know it, so you commit to the level of bombing deemed acceptable by the world at large. At best, you wind up in a situation like London during the Blitz. Your bombing runs are effective, in that they disrupt the daily life of citizenry, and cause some infrastructure damage and loss of life. However, you're never going to be allowed to scale up to the point where your victims feel they have no way out but to submit. There's enough plausible deniability that, even when a bomb hits close to home (literally or figuratively), the victim is more pissed at the bomber than their government.

The most successful besiegers were probably the Romans. It wasn't so much the act of laying siege that caused cities to surrender, it was the utter, uncompromising determination of the Romans to see the siege through to the end, and the atrocities they would commit on the surrendering population that made them so successful. Surrender immediately and you don't get enslaved or butchered... hold out and things will go very, very badly.

I don't recall all the details but there was one siege in western Europe where the mayor of the town declared 'you won't take us: we have supplies for four years in our store houses' to which the Roman commander replied 'then we'll take you on the fifth year.'

Or take Masada, a supposedly impregnable fortress built on a mountaintop. First the Romans built walls all the way around it, both to contain the Jewish 'rebels' but also to protect the Roman siegeworks from any potential rescue force. Then they just built a ramp. A massive, massive ramp, that reached all the way up to the fortress walls (which weren't that strong because who builds a strong wall when your fortress is perched on top of a mountain?). Then they wheeled up some siege engines, smashed their way through the walls and discovered most of the inhabitants had commited suicide rather than face capture.

Ye olde sieges cut off supply lines and forced the defenders to subsist on rations. Once those started running low, they started starving. Eventually the options were starve to death or surrender. These sieges frequently lasted months and sometimes years. Given travel times, it could also be weeks before anyone realized something was wrong and mobilized a force to break the siege.

Ukraine can only do infrequent drone raids. In order to properly siege Moscow, they would need to lock down all ways in and out of the city, and keep it that way for months, possibly longer given modern food preservation techniques and the viability of backyard farming. Additionally, sieging a city no longer prevents the people from communicating with the outside world, meaning other Russian forces would respond in days.

https://en.wikipedia.org/wiki/Soviet_invasion_of_Manchuria

Are you including Hiroshima and Nagasaki in that?

Those weren't prolonged grinding attacks on the civilian population.