You're right, the "most correct" answer is "it depends." But, let's sort through some hypotheticals... you mention that you're a governor, meaning that your town must be significant enough to warrant having one, and therefore probably has enough in the way of defenses to potentially make it annoying and costly to capture. You say you're surrendering peacefully - let's presume that mean that essentially the Mongols say "Hello, we're here, now submit." And your response is to immediately throw open the city gates and reply, "I thought you'd never ask, come in for some tea!" And finally, let's just go ahead and assume that you are not personally responsible for whatever reason the Mongols went to war against your kingdom in the first place.

You and your city are, more likely than not, going to be fine. You will face a tribute payment, yes. But especially if you're one of the "early birds" who crosses over before it has become "bleedingly obvious" that your kingdom is going to be on the losing side - and especially if you can get some of your kinsmen already on the Mongol side to vouch for you - you might be looking at fairly low levels of relative taxation. Let's take a look at a case study - the Mongol conquest of Khwarazmia:

Background

The Empire of Khwarazm existed across much of Persia, extending from the mountains of Aghanistan and Uzbekistan to the shores of the Black Sea, and including Transoxiana, Ferghana, Sistan, Khurasan, and Iraq-Ajemi. It was, in a word, huge. It was also a very young empire - having been won and formed in 1200 CE, it was only ~6 years older, in fact, than the Great Mongol Nation (Yeke Mongol Ulus) forged by Genghis Khan in 1206. In fact, across much of the first two decades of the 13th century, it was still far from a settled affair, and major fighting had only stopped in many of the regions as late as ca. 1215. As such the ruler of this Turko-Persian empire - variously styled as the Amir/Sultan, and self-styled the Shah (though the caliph in Baghdad rejected this presumption) - was every bit the brash conqueror-king with delusions of grandeur.

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The Mongol and Kwarazmian forces came to learn about one another existance right about this same time - the mid-1210's, albeit via radically differing means. The Amir of Kwarazm, having learned of a "Golden Empire" that he'd known as Kitay, far to the East that was wealthy beyond reckoning. Thus he dispatched an embassy/trade mission to go see what was up - since his men had informed him that there were disturbances in the region that threatened to undermine the lucrative supply of trade goods that transited his own territories. What they found after months of travel and entering the plain approaching the Central Capital of Jin, Zhongdu (mod. Beijing) was devastation beyond reckoning - a city so large as they'd never imagined with 40 foot high walls encompassing a region of more than 15 miles laid to absolute waste. The bleached, rotting bones of the city's dead stacked and strewn across the landscape in such great number that, at first, some in the party had mistaken the first sighting of whiteness coving the ground as an unseasonable snowfall... until the smell hit them. That was the first encounter of the Khwarazmians with the Mongols, or at least their handiwork.

For the Mongols of Genghis, it was a significantly less amazing affair. Having made war of the Kara-Khitan kingdom (AKA W. Liao), that had been usurped by an old enemy of the Great Khan - a Naiman called Guchlug - the Mongol pursue their defeated foe across the border and into Khwarazm territory, and they'd taken careful note of what they found. Genghis was greatly interested - a powerful state, and one far enough away to not pose an immediate threat, whose rulers had given him no cause for vengeance - in other words: a potential trading partner for the Silk Road goods he now controlled one terminus of.

After a round of negotiations, a deal was struck, and the Great Khan dispatched a vast caravan loaded with every manner of luxury - camel furs, silks, gold, silver, tea... you name it. And some 450-500 merchants and traders loyal to the Mongols, though primarily Muslim, and foreran by an Indian emissary bearing a letter of friendship to Amir Ala al-Din Muhammad II:

​I have the greatest desire to live in peace with you. I shall look on you as my son. For your part, you are not unaware that I have conquered North China and subject all the tribes of the north. You know that my country is an ant heap of warriors, a mine of silver, and that I have no need to covet other dominions. We have an equal interest in fostering trade between our subjects. [... I invite trade between our two peoples so that] henceforth the abscess of evil thoughts may be lanced by the improvement of relations and agreement between us, and the pus of sedition and rebellion removed.

It was not, however, to be. Fatefully, the Mongol caravan’s route passed through the Khwarazmian province of Syr Darya, and its regional capital Otrar. The governor of Otrar was Inalchuq, a relation of Amir Muhammad’s mother. It was he who stopped the caravan’s procession and, after meeting with their ambassador and the othe heads of the group, order them all detained. The stated reason was that one of their number – a man Inalchuq had known previously – had supposedly addressed the Governor without his title, Ghayir-Khan. It would become quickly obvious, however, that this was little more than a convenient excuse, and that the governor in fact simply meant to seize the caravan’s rich cargo for himself. Upon taking the Mongol caravan captive, Inalchuq send a message to his Amir, telling him that he had captured a host of Mongol spies, and proposing their immediate execution. Whether because he actually believed his governor’s accusations, or – rather more likely – that he saw this as a chance to seize the goods for himself without need to pay it back, the Amir agreed to the execution of the entire party, and that the stolen goods be taken to Bukhara and sold.

In spite of this egregious offense – the killing of a Mongol ambassador was one of the most serious possible offenses, as was the theft of the Khan’s goods – Genghis as yet was willing to make a final attempt to avoid all-out war. A new embassy was sent to the Khwarazmian capital, this time consisting only of one Muslim and two Mongols to demand an explanation from Amir Muhammad for this egregious breach of international etiquette. Perhaps, after all, the Amir was blameless, and the blood lay on the offending governor’s hands alone. If that were the case, the ambassadors demanded, Muhammad must turn over the governor of Otrar to them in chains, to be taken back to Mongolia and face the Khan’s justice. At this, Muhammad balked – he would do no such thing, and the fact that these three foreigners had the audacity to stand before him making such demands was an insult to his station and power. Especially insulting to him was the fact that the lead ambassador was a Muslim – like himself – yet paid his service and loyalty to this heretical barbarian king. Muhammad II’s response was swift and damning: the Muslim ambassador was executed on the spot, and the two Mongols were instructed to ride back to Mongolia and inform their king that if he wanted his treasure back so badly, he’d have to come and take it, if he could. Yet before they were allowed to leave with their lives, the Amir had their beards shaved off – a sign of extreme humiliation – and the executed ambassador’s head sent with them in a box. All this was in 1218... the Mongols rode out for Khwarazm in early 1219. They'd arrive that autumn.

From the Secret History of the Mongols:

[Genghis] rode past the settlement of Alai, taking with him his Merkid wife Khulan Khatun as companion, and leaving his brother Ochigin in command of the Great Camp at Avarga. The first army he sent out was commanded by Jebe, behind them he sent another led by Subedei, and behind them a third under Tokhuchar. Sending off these three commanders he told them: ‘Ride off to the outside of the Sultan’s armies and wait them until I arrive. Then you’ll attack from your side when I strike from mine.’

Though at first the Mongol armies rode past Kwarazmian cities without stopping or attacking, their mere presence had the – perhaps foreknown and intentional – effect of breaking most of the populace’s will to put up any resistance. In droves, the populations fled to the Mongol lines to submit, and bringing with them yet further information about the fracture-points within the military and political situation of the enemy empire. Elsewise, they fled in the opposite direction – toward the nearest great city of the region: Bukhara. As far as the Mongol commanders were concerned, that was actually the better outcome. Just as they had done to the Jin Empire in the east, they now used terror as a means to stretch the supply limits of this sedentary society to, and then past, its breaking point. If they would hide behind their walls, as the Jurchen had done and as these Persians surely would as well… well, then best to fill those walls to the brim with useless mouths and grasping hands. Better to deplete the cities’ supplies and pillage their fars well ahead of the first mongol arriving at the city gates. The border towns under the command of Amin al-Mulk were thus taken and plundered, and the governor himself fled to the prince Jalal al-Din in an attempt to bolster each other’s forces against the Mongol advance.

OK... so now having established the relevant groundwork, let's get into the meat of your question...

This was an incredible and fascinating read! People mentioned a podcast? WHERE?!