Dry ice doesn’t melt, going directly from solid to gas. The powers of D&D casters, similarly, sublimate. One day they have no access to a spell. The next, its casting becomes commonplace. This is most obvious with the spells that drastically change the capabilities available to the party: Fly, Teleport, major Divinations, the Wall spells, Stone Shape/Move Earth, Fabricate, etc. However, many other spells roll in with each level that are a sea change beyond what was capable before. A single level’s difference frequently makes a huge difference to how a party will tackle any given threat, unless arbitrary and obvious countermagics are built into the scenario precisely to make spells useless that could circumvent the design.
But the problem is not that casters dramatically increase in power with each spell level so much as it is that this shift is not at all gradual. Between spell level 4 and 5, a party with a wizard can change their expectations between taking weeks to reach far-flung allies to popping in to visit them every day via Teleport. Between 2 and 3, challenges with a Z-axis go from a big deal to negligible with the addition of Fly. The issue is that, if a caster can cast a spell at all, he or she can likely cast the spell multiple times per day, every day. The impossible becomes commonplace with no transitory state.
The simplest solution to this is to give out more scrolls of higher-level spells, but this has some drawbacks. Primarily, a scroll isn’t just an ability to cast a higher-level spell, it’s adding the spell to the caster’s repertoire whenever it’s needed. Unless scrolls are arbitrarily given out knowing they’ll be needed in the scenario soon, the GM risks a player hoarding them and unloading them all at once to trivialize content, rather than to instill the desired sense of growing power.
Another solution is to create a mechanic whereby casters can ritualize certain spells that they ordinarily are too low-level to cast by expending several spell slots, additional time, and some form of non-expendable spell scroll. But, again, this may require very careful playtesting to find out what’s unbalanced. Some higher level spells may be so good that casters will regularly take the extra time and slots to use them as rituals, thus not really solving the problem.
The best solution, for the purpose of creating a “liquid” state of new spells may be to create a rote mechanic whereby newly learned spells cannot be fully utilized until they are gradually improved. This might mean that, for example, Teleport has a cooldown and/or a distance limit that is gradually expanded to its normal numbers as it is used. However, this solution is, unarguably, a nerf to casters, and would likely need some kind of commensurate bonus to account for it.