Greetings! This is a rather broad question to deal with, but it is an interesting one on the maintenance of imperial rule nonetheless. I'll be adding on to u/DarkJayBR's great answer with some key "considerations" if you will for a fantasy novel (best of luck with that venture by the way!). Obviously my knowledge here is limited mostly to the British Empire and the other European empires of the 19th century, so should you dear reader (yes you!) have some nugget of gold about the rapid redeployment of troops in other Empires across the past, feel free to add onto this thread by all means! Let's begin.

First and foremost, we have to consider the nature and size of the conflict itself. OP has stated in their explanation that this is for a "large conflict", but that again depends on the scale of the empire itself. For instance, are we talking about a provincial, regional, or even cross-regional conflict breaking out at the fringes of the Empire? For that matter, what is the actual nature of the conflict? An empire dealing with internal revolt and rebellion is a wholly different matter to full-scale war with an external rival or challenger. The scale of the conflict is important again in this case. Is the empire having to deal with a threat that is able to constantly wage a sustained campaign against it (i.e force back the troops of the empire with the intention of conquering its territory), or is this a particular settlement that's decided to carry out the occasional skirmish against an imperial outpost in the region? Depending on how wide the conflict it, the nature of the opponents, and the intention of its combatants, that will often dictate how many troops an empire will call from other colonies to deal with it.

Case in point, the British Empire in South Africa had to deal with skirmishing bands of boertrekkers and the local tribes (namely the Xhosa and Nguni) throughout the late 18th and early-to-late 19th century. But because the nature of warfare out on the open veld (the plains of the South African landscape) favoured hit-and-run attacks, only a police force staffed by British officers and local recruits was required to fend off the nuisances posed by these internal "dissidents". On the other side of the spectrum, when the Indian Rebellion of 1857 (otherwise known as the Sepoy Mutiny) broke out, the sheer scale of the conflict (spreading across the subcontinent and sweeping through many provinces) meant that the British had to dispatch troops from the Home Isles to prop up the local armed forces.

The second factor which links nicely into the previously one is the simple geographical location of the empire's holdings in relation to the conflict. For many European empires in the 19th century, key ports, capital administrative towns, and even provinces contained a barracks of some sort. That way, a localised revolt could be quelled relatively easily, without the need to appeal to the "mother-country" for reinforcements (in the case of the British Empire that meant appealing to Whitehall for military aid). If however, an external conflict arose in an area where the empire does not possess holdings, it is entirely possible that those troops could then serve as the contingent to fight for the empire in that war. A notable example of this was the First Opium War, in which the British government decided to raise a "punitive expeditionary force" against the Qing empire. To do so, they called on Royal Navy assets from as far afield as South Africa, Australia, the Mediterranean, and even those still under construction in the Home Isles (they were to rendezvous at British Singapore, alongside regiments of the British Army and the East India Company's Army). In other words, it's more likely that troops will not be called from as far away as "the other end of the empire", since presumably (though not always), there will be armies (or the means to raise one) stationed closer to the area of conflict.

Finally, this is a less relevant factor in a fantasy novel, but the timeframe and technology is also a key consideration for how an empire would deal with conflict. In an age of sail and volley fire, the deployment of armies from one end to another could take weeks if not months, and that might in turn result in the conflict escalating in size and urgency. Returning to the First Opium War, although the plans for the British expeditionary force and the process of gathering it had been enacted since January 1840, the actual troopships and troops themselves would not see combat until mid-June, simply due to the vast distance from the rendezvous point to the conflict area.

Hope this bit of a response helps, and feel free to ask any questions on what has been mentioned here!

Depends on the Empire and depends on the situation. Some empires like Rome, Macedonia and the British usually assigned governors for all conquered provinces and these governors would have access to fixed Legions/Troops nearby that would be under their command to put down rebellions, thiefs or invaders. Only if the situation got out of control, that the King/Consul/General would march down from the capital to the province with the main army to deal with the issue.

Some empires like the French usually didn't leave stationed troops on conquered countries. So if they rebelled again - the Emperor/General would just march down there to put down the rebellion with the main army.

Responding your second question, yes, it was also common to pull your army from the other end of the Empire to deal with a crisis. Julius Caesar pulled some Legions who were dealing with the Dacians in Croatia to deal with the Galls on France. After the Civil Wars were over, he planned to go back there and put the Dacians on their place, but he was killed in the Idles of March before he could do that.