Standard rules: after being established once at the start, initiative order does not change during combat.
It's not the rolls that you keep but the initiative order as a whole. Further the rules state that a dexterity check to establish the initiative order occurs at the start of combat and that this order remains the same throughout combat and is not updated.
Thus initiative values ( = results of the dexterity check) do not change even if dex modifiers change. To be more precise: they do not even formally exist any more during combat. They're used for establishing the initiative order and then discarded.
From the 5E Basic Rules, emphases mine:
Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time.
The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.
However, there is a variant presented in the DMG that changes this.
"Speed Factor Initiative," presented on DMG pp.270-271 requires a new initiative roll each round, and so would be responsive to changes in modifiers. AngryGM writes about it a bit here (warning: profane and vulgar language punctuates excellent analysis and description), and you should be aware of some of the issues raised in these questions (1, 2, 3).