Donna, one of the mistakes I often see beginner quilters make in lattice and borders, is not measuring first. By just laying a quilt or a block on a long strip with the intention to cut off the leftover, never works out well. One of the problems is that the feed dogs are pulling the underneath fabric. This can lead to rippled borders, or in your case a puffyness where you don't want it.
There is a simple remedy for this problem. In the case of borders, always measue the center of the quilt and cut borders to that measurement. The outside edge of the quilt top has too many seams to allow for an accurate measurement. Fold the resulting border strip in half and mark the center with a pin. Do the same for the quilt top and pin first there. Next, pin the end of border strip to each end of top. On a bed quilt you would proceed to find the middle between the two pins and continue in that manner. Only sew on the border once it is evenly pinned to the top.
Lattice would be done much the same way. Find the measurement of the block, cut the lattice strip that measurement, pin middle and ends. If your quilt has a long lattice strip between rows, make sure to take the measurement in the middle of the row and not at the edge, for the same reason prev. stated. If you are using cornerstones, the pressing is the key to success. Cut all your lattice the same length as your block, make one long strip with the cornerstones inserted. If you have pressed your side lattice away from block, then make sure to press away from the cornerstones to allow the seam allowances to lock.
I am sure if you follow these guidelines, you will have much better success in obtaining a nice flat top without the puffyness that is currently giving you fits. Good luck to you, and please consider posting a picture of your completed project. We love seeing others work.