次第しだいに

つつある goes on a verb's end, saying the change is happening now (増ふえつつある). 次第しだいに stands on its own, and can describe a finished change (次第しだいに暗くらくなった).

だんだん is the spoken word, reading chatty in writing. 次第しだいに is the written one, stiff in conversation, yet だんだん survives in keigo (だんだん分わかってまいりました).

次第しだいに just says a change happens step by step (次第しだいに暑あつくなる). ますます says a trend already running has picked up (ますます暑あつくなる).

次第しだい rides a ます-stem, and fires the moment something finishes (届とどき次第しだい). 次第しだいに stands free, describing a change with no trigger (次第しだいに暗くらくなった).

次第しだいで follows a noun, making the outcome depend on it (天気てんき次第しだいだ). 次第しだいに describes a change instead, with nothing to depend on (次第しだいに暗くらくなった).

An i-adjective has no verb to modify, so 次第しだいに uses くなる instead (次第しだいに暑あつくなる). Never ×次第しだいに暑あつい alone.