The pattern ～てくる is also really useful way to express a change in state. I remember it being a confusing pattern when I first started studying Japanese, but I solved that problem by using it constantly (regardless of whether or not it was appropriate) because, let’s face it, shit be changing, yo. (Cue Tupac track.) Gradually I’ve narrowed down the appropriate usage conditions.
Here’s a quick example sentence: この2ヶ月間仕事が忙しくて大変だったけど、やっと落ち着いてきた。
The first clause sets the scene – work was busy and it sucked. But (there’s the fulcrum for the change) things have settled down (or so he thinks). The second clause shows how things have changed from the first clause. They have gone from an 落ち着いていない condition to an 落ち着いている condition. The きた, I feel, helps emphasize this transformation and the やっと impresses exactly how long the subject was waiting for the きた to くる so he could キター！
So the next question is what’s the difference between ～た and ～てきた? It’s pretty subtle. Another example. I got a birthday package from a friend in the US the other day. If my friend spoke Japanese I could say that it got here with a simple 荷物が届きましたよ. Just the basic way to say that the package was delivered. I could also say 届いてきたよ, which feels more urgent – as if my friend or I were awaiting the arrival of the package slightly more desperately. Maybe the package was delayed? Maybe there was a kitten inside?! Or maybe we were just waiting to compare the USPS with Japan Post. (Japan Post wins.) As in the example yesterday, the きた・くる seems to address a sort of mutual understanding that exists whereas the plain form of the verb is best for communicating brand new information.
〜なってきた・くる is another pattern that you can use to express how something changes. An example: 寒くなってきた. Like Monday, the timeframe feels more specific, as in 最近 or 今日 or 今週. And once it’s cold enough, (you can use Monday’s pattern to say…) 雪が降ってくる.