I've actually never thought of it running into problems. Being moved around appointments isn't limited to Japan after all.

国替 Kunigae, literally "switching country", is the historical term for it, and was fairly common in the Sengoku and early Edo. By most accounts the daimyō being moved would uproot himself and all his vassals to the new location, and they would be assigned new locations. His vassals would assign their vassals, and so on. How low this went seemed to have depended case-by-case. Ieyasu seem to have at least assigned from the most high ranking of his vassals, the upstart Ii Naomasa assigned 120,000 koku after the move, all the way down to Ieyasu's personal retainers at the 200 koku mark. When the Uesugi moved to Aizuwakamatsu though, it seems they even took with them a considerable amount of peasants. This became part of the sticking point between the Uesugi and the Hori who moved into Echigo that the Uesugi vacated, though this could be apocryphal.

Possibly apocryphal but is from later official Tokugawa sources (the Tōshōgū Gojikki), Ieyasu's vassals did complain to him about being moved far away. Other stories of similar complaints about getting moved, though many are known to be apocryphal, does leave the impression that not everyone was happy about being moved, even if it means to richer pastures. And while I don't believe this is the case due to the nature of the source, it is possible that one of the reasons Akechi Mitsuhide turned on Oda Nobunaga was because Mitsuhide was going to get moved.

To help transition the move, it seems there were some sort of arrangement that was supposed to be followed. I can not find anything that would tell me whether it was an actual rule, an unwritten rule, or just an ad-hoc arrangement, unfortunately. But what was supposed to happen was that either you left taxes (rice) uncollected, or if already collected you left a significant portion (half?) of it behind when you moved on, so the guy moving in has something. There are also a couple of (possibly apocryphal again) stories of people taking more than they are supposed to when they left though, including the Uesugi which might have contributed to the outbreak of the Sekigahara campaign. Presumably, Ieyasu didn't need to worry about the Hōjō taking all the tax revenue of that year because the Hōjō were not being moved, they were destroyed.

As for Hōjō's vassals, they became rōnin. We know some found employment under Ieyasu, others Hideyoshi or some other daimyō. I, obviously, can not say they all did, and I suspect many didn't. Those of the Hōjō clan that were not ordered to commit seppuku were forced into monasteries to live out their lives as monks.

At the time, the Hojo clan was besieged by the armies of Hideyoshi. It was during the siege that Hideyoshi proposed to Ieyasu the trade of the Hojo's provinces for Tokugawa. Ieyasu's provinces were Mikawa, Tōtōmi, Suruga, Shinano, and Kai, which were located very close to central Kansai region (home of Kyoto) and could potentially threatened Hideyoshi's position for power. Hideyoshi was distrustful of Ieyasu as they had a few disagreement in the years since they fought as allies. The trade therefore gives Hideyoshi a chance to push Ieyasu aside from any position to challenge his authority. Ieyasu accepted the trade and once the Hojo surrended, he owns the Hojo's land and moved his own soldiers and vassals to pacify any Kanto samurai who were uncertain of how he would rule.

In the end it worked out well, as Ieyasu reformed the provinces and improved the economic infrastructure of the land. The relative distances from Kyoto allows Ieyasu some freedom to rule and enhance his own prestige while Hideyoshi struggles to maintain his own legitimacy to power in Kyoto.