gradually; little by little; step by step
Describes a slow, steady change over time. Often used with verbs of change like なる, 変わる, 増える, 減る, etc.
The sky gradually grew dark.
His health is gradually recovering.
The sky gradually became dark.
I gradually started to understand what he was saying.
The population is gradually decreasing.
I gradually got used to my new life.
次第に is a broader, conversational word for gradually or as time goes on, whereas 漸次 is limited to formal or written contexts.
次第に describes a gradual progression like 日に日に, but it doesn't carry the day-by-day imagery and can describe change over any timescale.
次第に means 'gradually' over any timeframe, whereas 日増しに specifically marks change that accrues day by day.