I’m sorry, I thought what I was giving you was about catching in floats in color work which is what I thought they may be talking about in the directions. Here is a link that really goes to something about that. Link Scroll down to where it talks about “wrapping the yarn while you knit”.
Having said that, it may not be what they are talking about and I don’t think I would use it in this situation. I tried it out on a sample and you have to put these changes on top of each other and it specifically says to avoid that or it will show through and it does.
I looked at the mittens again and most of the changes are just one or two rows. For those I don’t think I would do anything special. I’d just start knitting with the new color being careful that there is enough slack where you pulled the yarn up to use it that the stitches aren’t distorted. If that didn’t make a good result I would do what I tell about below.
Another way you can carry yarn along in this case is at each round change to run the working yarn under the waiting yarn as you knit the first stitch and it will catch it along. I don’t like that much, especially over long distances but it is often recommended. You could try it. I find it leaves the row change area not looking so nice.
On the mittens you have there is only one place (I think) where you have to carry a yarn a long ways, and that is when you start using the gray again after the wide section of orange.
Here is how I would handle that section if I were doing it. I would just cut the gray off after I finished with it and work it’s tail in later. And start working with the orange. Then on the last row of the orange about 2 inches before the round change, hold the gray yarn up next to the needle and work over it. Here’s how. Hold the tail end of the gray yarn off to the right and the yarn that goes to the ball right behind the left hand needle (LHN), next to it. Leave about 2 inches of gray tail out beyond where you begin stitching. Now as you work with the orange when you insert the needle into the next stitch make sure your needle goes under the gray yarn, don’t catch it in or anything, just run the right hand needle (RHN) under it and knit the orange stitch. Then on the next stitch make sure the RHN goes over the gray yarn as you knit the orange stitch. Alternate knitting over and under the gray yarn until you come to the round change. (This will knit the yarn in without you having to work the tail in later.) Then switch places with the yarns.
Take the orange yarn and stretch it out next to the LHN and with the gray knit under and over it for a couple of inches. Then before you cut the orange off, pull on the tail that goes to the ball to even out the tension of the carried yarn. (do that with the gray above before you cut it flush as well) Then you can cut it off. I am much happier with doing it this way than trying to carry the yarn a long distance and having it look all wonky at the joins.
If when you look at the inside of the mitten if you just knit from row to row on the narrow row changes you find you don’t like the look of the strands that were pulled up to knit with you can tack them down by sewing them to the inside (don’t work all the way through, but divide the yarn you are sewing into).
I hope this is of some help.