Now, moving profile pics are very popular. You can make your own using a picture editor such as gimp or photoshop, though it is complicated so I will tell you how to find good ones for your profile pic, etc on the internet.

Above is an example moving picture I found on the internet. I found this on google.com, but you can also go to deviantart.com and search gif or moving pictures in the searchbar.

Searching for moving pics on google images-
You can type a few things in the searchbar. Here is a default search you can type-
Gif (topic you want your pic about)
or
moving picture animation of (topic you want your pic about)
I prefer the first one, much better results will come. 
Many pictures will not work for your profile pic though. They might exceed requirements, that will be the message that appears when they don't work. You have to make sure the pic is square!

Thanks and good luck finding a pic! post any questions or comments below. Here are a few examples for you to see :D