Another relevant point is that the nomads were unlikely to have developed significant logistics chains. Being nomads and herdsmen, they are highly efficient when they are roaming over good grazing country for their mounts and herds, and can move extremely fast and require little supply. But as soon as they DON'T have that, they are forced to forage for grazing/provender and their own supply. Going on campaign like this, they're not bringing their herds with them, those stay with their clan. It's their wealth and means of sustenance, it's too risky to bring on campaign. So it's just the nomads serving as warriors and whatever supply train/herds they've brought. They intend to raid and forage for everything else they need, possibly rely on the commander they're serving to provide some supplies as well.

But foraging and raiding have diminishing returns. You can only rob people once, twice if you were generous the first time or waited awhile. Once you've taken everything, there's nothing left. Raiders drive sedentary people away, and they won't come back until they're gone. No point in going home and planting again, just to get raped and pillaged again, if you have any other option. Armies also hunt more game than is sustainable; armies are like swarms of locusts, they strip everything bare. Which means if you're trying to live by foraging (that's not the robbery kind of foraging) you have to keep going further and further afield to find anything, which goes the same for raiding. The further out you are, the more isolated you are, the more likely it is you encounter a superior enemy force and get destroyed. The more the army disperses it's strength, the less effective the siege is. Messengers slip through the lines, the defender sally out, relief forces can get closer without being detected by scouts.

And finally, nobody wants to be the attacker in a siege. Siege camps are awful, filthy places where disease is rife. You're sitting there every day, digging and working and getting shot at, with very little ability to strike back. It sucks, and it's hard to motivate an army to do it with gusto. Nomads are even less likely to care; they are here to raid, it's what they're good at and what they want to be doing, for the reasons above. Fighting in a siege is the kind of thing they absolutely don't want to be bogged down in. Sieges also drag on and on, and most end without taking the city or bastion. It has a low chance of working, honestly. It's simple math to see what a bad deal it is for the nomad raiders, something they are keenly aware of.

There's also the nature of the nomad clans and their relationship with the commander. Nomads are, universally, proud and independent people, shaped by their way of life. If they don't like you and your bullshit, they pack up and move on. They aren't feudal serfs, who are tied by land, oaths, and family back on that land to their lords. They have a high probability of telling you to fuck off and riding away and you not being able to do much about it. They are less levied troops or sworn soldiers than allies of convenience. They are here for the raiding and booty, and your campaign is a good excuse for it. As soon as the raiding and booty prospects look bad, they're gonna fuck off and find better prospects.