For example, a character takes the Magic Initiate feat at level 4, picking spells from the Wizard spell list, because they have been researching and learning all they can about magic (such as a Sorcerer studying their inborn magic or a High Elf expanding on their Cantrip feature). It stands to reason that they would continue their research of magic over their next 16 levels, which indicates that they should learn new spells.
At any point, can the spells chosen through the Magic Initiate feat be swapped out for other spells, according to the same Magic Initiate spell selection rules of two cantrips and one 1st-level spell?
Best Answer
You can't automatically replace the Magic Initiate spells later on
If a rule allows for retraining it would say so. Magic Initiate does not have any such rule.
Continued study would be simulated by Multiclassing
Whatever character your playing is progressing as a main class, and most of their attention is dedicated to that progression. The Magic Initiate feat captures a small amount of attention devoted to another class, but continued study (such that you would acquire new spells) would be emblematic of a multiclass wizard.
Spellcasting
(this section is pending revision upon further debate here)
If you gain a level in a class that has Known Spells for which you also have your Magic Initiate spells, you could use the Spellcasting feature to replace it:
These spells still count as [class] spells since they appear on that class's spell list. As such, it is eligible for replacement by the Spellcasting feature, but will still be subject to the rules that: