随之【suízhī】

随之 says that something follows along with a change just mentioned: "along with it", "accordingly", "in turn". 之 is a written "it" pointing back at the first clause, so 随之 means "following that". It goes before the verb of the second clause, often after 也.

The verb after 随之 is usually one of change: 增加, 提高, 下降, 改变, 调整, 消失, 产生.

随之而来

随之而来 means "comes along with it", for a consequence that arrives with a development.

随之 or 随着

随着 takes a noun phrase and opens the sentence: 随着经济的发展 "as the economy develops". 随之 takes nothing after it – its "it" is already built in – and it sits inside the second clause. The two often appear together.

• × 随之经济的发展，生活水平提高了 → 随着经济的发展，生活水平提高了

It is neutral to written; in speech, 跟着 often does the same job.