てきた help

I am rather confused with the usage of てきた. I'm referring to the one that's not the used with physical movement. Some examples:
話がだんだん面白くなってきた。
何となく分かってきた気がする。
すごく太ってきた。
やっと最近暖かくなってきたよ。

I understand how てくる/ていく work (will happen/become, etc.), but てきた is really confusing me. I looked up a lot of examples on the net like the ones above. From what I've gathered, てきた is supposed to be "present perfect" in English, kind of. While the difference between なった and なってきた is that the latter happens gradually. I'm not really sure about it, though.

I know I shouldn't rely on the English translations all that much, but like all of the example sentences on alc with なってきた are translated as "getting" or "starting". Like this example - やっと最近暖かくなってきたよ。(Finally, it's getting warmer lately.). I would expect to see てきている there. From what I know てきている should mean that this verb "keeps happening". I have no idea what's going on here. :/

I would really appreciate if someone could give some clarification on this てきた grammar point.

The key is that ～てきた expresses a continuous action/movement in the time line (mostly from past to the present). It is not used only for "gradually". You can also say 最近急に暖かくなってきた. ～てきている means that the movement is still ongoing, since ～ている form expresses present progressive here. For instance, 暖かくなってきている implies that tomorrow will be warmer than today. Your understanding "keeps happening" is not wrong in this sense.

Thanks, as always.
Edit: Just wondering what the difference between 最近急に暖かくなってきた and 最近急に暖かくなった would be? The former describes the present and the latter past?

最近 shows that both are about the present.
暖かくなった expresses the present state after the change, whereas 暖かくなってきた focuses more on the change itself. なる expresses the change by the verb itself, so the difference between the two expressions is subtle in this case, though. (For instance, "太った vs 太ってきた" more clearly shows the difference.)

I see, thanks.