For a more specific example consider this. This is a campaign I documented for my blog in 2013. I am not going to go blow by blow but highlight the unforeseen changes in the campaign that resulted from the players deciding to go right instead of left.
For more details see this pair of blog post. And yes I am guilty of not finishing part 3.
Part 1
Part 2
The campaign starts out in A. The players decided to start out as a mercenary company. I pitched several "contracts" to start out with. The one they opted to go with is working for the Baron of Abberset patrolling the Estoil Hill that was the southern frontier of the Principality of Nomar.
The cast of Characters (created with GURPS)
Delvin - a dwarven fighter from Thunderhold wandering human lands to seek his fortune.
Durgo - a forester from southern Nomar who left his home and village under bad circumstances
Sir Cei Kerac - a hedge knight hiring on as a mercenary to regain the fortune he lost gambling
Aeron - a ordinary common man with special skills
Kermit - a half Viridian (demon), half Karian (Japanese) magic using puppeteer.
Sir Henry Kiefer - a young knight seeking to regain the manor that is rightfully his.
Brom - A Warrior seeking his fortune.
After the first few sessions they learned the following in the linked PDF. This was created at the players' request to summarized what they learned.
Summary of the Halkmenan Rebels info
The first left instead of right change occurred between D & E. The players used a map they captured to find a dungeon (it was one level). Explored it and found the treasure. Sir Cei Kerac's players convinced the group that it would be more lucrative to move north and sign up with the Prince of Dorn (the ruling prince of the country) who was hiring mercenaries to fight the Skandian Vikings of Ossary along the frontier. Think of the prince as a version of King Arthur or King Alfred, and the Skandian Vikings like Saxons or Danes. Not quite a all out war but not quite peace either. Most of the treasure was used to buy out their contract with the Baron of Abberset. Who wasn't happy but the party did everything properly and legally so that was that for the Baron.
This decision totally trashed a lot of my prep but as usual I rolled with it and used my Bag of Stuff and some random tables to handle the journey north. I used the following two weeks to prep what was to happen when the party reached 'I' on the map.
The next phase went well and due to the party's attention to detail wound up with them capturing the King of Ossary. This is was the point where Sir Cei Kerac's player couldn't continue with the game and had to leave. The party received an substantial offer from the Prince of Dorn to purchase the king's ransom from the party. The party then traveled to the capital of Dorn and purchased land on the crossroads at N in order to build an inn.
The capture of the king was unexpected. They figured out the king's schedule of hunts, patrols, and visits and acted accordingly. But the king had a substantial guard with him so the outcome wasn't certain. The most likely I thought was the party giving the king an ass-beating forcing him to retreat and losing many of his personal companions. But they bagged him alive and in doing so realize they had a literal king's ransom in their hands.
As Sir Cei's player was the driving force behind being mercenary and more importantly working this particular spot in the world. The party felt no obligation to stay there after he left. So they debated their future and settled on building an inn. Figuring asking for the land as a freehold, it was a feudal society, was a reasonable ask. Then using the reward as capital.
So this required me be to do another round of prep centered on the crossroads at H. Again I had about two weeks as the players went Dorn got their charter, and then went to H. The rest of the campaign was spent dealing with the situation around the crossroads. Namely dealing with the Baron of Oxcross who was offended at the party getting a freehold. And the problems of the Plain of Cairns to the east. Mostly exploring barrows, fighting undead, and winding up with breaking the curse of the cairns.
After the curse was broken, the inn was built and that where the campaign ended as it felt like a natural stopping point.
Rob's Notes
I hope this illustrate what people do in sandbox campaign. It started with the question to the group "What would everyone like to play?". They threw out an idea, mercenaries. I pitched several situations. They picked Nomar and Abberset. I then set down to prep it. Because they were employed mercenaries they were subject to orders, which means missions. Which most at first glance would not consider to be a sandbox.
However I didn't consider the campaign premise to rest on the fact they were mercenaries in the employ of the Baron of Abberset. As long as you are willing to abide by the consequences, don't worry about what I have to do. Go left instead of right if that makes sense to you. Which why the party decided to pay off the Baron of Abberset as they didn't want that millstone hanging around their necks. They knew if they just left that eventually in the next few months of game time that the Baron would catch up to them and there would be hell to pay. And given that this a feudal society and the best the party can muster socially were knights. They would be fucked especially the knights who may lose their title and privileges as a result of breaking a legal contract.
The same with the transition from the second to third phase (inn building) of the campaign. Except the players knew their fate was in their hands given the opportunity that laid before them, the capture and ransoming of a king. So they were much more proactive about this compared to lucky find of a dungeon and its treasure the previous time around.
Mostly of my campaigns are like this. Slow development of the player's goals followed by a sudden change in the course of the campaign as result of circumstances or deliberate decisions. Yes a small number of these campaign amounted crawling from hex to hex when the player wanted to explore and region. For example the orc infested Forest of Dearthwood next to the City State of the Invincible Overlord.