as 与 with 都有“随着……”的意思吧.

当as 引导一个从句且位于句首时,应注意弄清 as 是表示“原因”还是表示“与…同时”.句子As they were leaving,I walked to the door 既可表示我向门口走去,因为他们离开了,也可表示在他们离开的时候,我向门口走去.
At the same time that; while:
同时；当…时：
slipped on the ice as I ran home.
我跑回家时在冰上滑了一跤
随着 [简明汉英词典]
along with
随着,对应
An independent man,Peter never goes with the tide.
彼得是个有主见的人,从不随大流.
顺着,沿着（某个方向）
to sail with the wind
顺风航行

as
adv.
同样地, 被看作, 象
prep.
当做
conj.
与...一样, 当...之时, 象, 因为
with
prep.
有, 以, 用, 同...,由于, 和...一致, 赞成, 关于, 就

的确都有"随着"的意思
e.g. as the year went by
with the time going
都表示"随着时间的流逝..."