彼は徐々に歩けるようになってきた。空が次第に明るくなってきた。 these two words are mainly used in written language, and sound a little bit difficult. In this context, i think 少しずつ　or だんだん　is more common and often used. 彼は少しずつ歩けるようになってきた。空がだんだん明るくなってきた。also, you can use them in conversation. ぼちぼち is little by little, but it's used when someone does something, like "i will do it little by little" and this is not so common, i've never said this word. As for 漸次,、i rarely see this word.