
Step 1: Create a new image. The size doesnt really matter, I use 450x100.
Step 2: Go to Filters>Render>Clouds>Difference Clouds. You can change the settings to whatever you want. Im leaving the settings as is. That will leave you with this:

Step 3: Duplicate the cloud layer. Then move the copy layer to the right until the left side of the copy layer touches the right side of the original layer (Yes, it will not be seen in the image). Then merge down so there is once again, one layer. Now you should have this:

Step 4: Move the layer over so you can see the line that is in between the two sections.

Select the smudge tool and smudge across the line going from the lighter regions to darker until the line cannot be seen.

Then select the blur tool, and set the rate to 100. Run it over all of the harsh edges to make it blend.

Now move the layer back to its original position like so:

Edit:An easier way to do the above comple steps that I didnt realise is to either use solid noise and make tileable or do Filters>Map>Make Seamless.
Step 5: If you have GAP (GIMP Animation Package) then there is probably an easier way to do this, but if you dont, read the rest. Duplicate the layer and move it a tiny bit over. You will repeat this process a lot of times to pick an increment that you can remember. I chose to do this:


This will decide how fast it will move across the screen. A smaller increment will make it go slower while a bigger increment will make it go faster.
This will leave you with the finished product I already showed you. If you want to put something behind the fog like this:

Then you have to lower the opacity lower and then put what you want behind it under each of the fog frames and then merge them.
Edit: A suggestion by Oregonian was that you can change all of the frames to Grain Merge instead of lowering the opacity.

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