Does skill transmit to other time controls ?

So recently, I started playing mostly rapid, while I was only or almost playing blitz. My blitz rating was plateauing around 1600, and my peak was 1658. My w/d/l rate was very equal (46/8/46). So I started rapid where my rating was much lower (~1200). I had some problems adapting to the time control (mostly taking my time to analyse deeper), but when I got the hang of it, I quickly climbed up, and I am currently on a little win streak (8) and my rating is 1420. But my question is : do you think my improvement would translate to blitz, or is it just because my blitz rating was much higher ? Because I feel like I'm improving, but it might just be because I am adapting to rapid and that is the reason why my rating has gone up. 

Like is my rapid rating climbing because I am just better than what it was when I came back, or is it really improving. Am I just getting to the rati g that my skill represents or is my chess getting better ?

Your 1200 rapid rating was probably low. Rating should plateau as your win rate and loss rate equalize.

Which has been the case in Blitz in fact! But my question (badly asked I'll admit) is will skill from rapid (board vision, finding tactics), things you don't have the time to do in Blitz. Will I be able to do these more easily ?

Thanks for your first response though !

Ratings at different time controls are correlated. However, the slower the time control, the more accurately the rating reflects chess skill. At faster time controls, there are more “nonchess” factors that affect game results, such as quality of the internet connection, speed of one’s computer, and quality of the mouse

Even so, there are some people who are especially fast or slow, which may make their ratings at different time controls more variable than those of others.

It's all chess. I find that improvement comes from studying key positions from a game, not playing the game. Any time control can be used for this if you are able to remember a key position and your thought process (so maybe not the fastest time controls where there are few or no thinking moves - I have never played a game faster than 5+0).

I know people that play nothing but online speed chess and they are quite good at it. But none of them have gone past USCF C class, so they prefer playing speed chess, once they found out that the "tricks" they use online don't work OTB in slow time controls. But as with most things in life you need to build on the basics first. When you learned to read you didn't start by speed reading.