(1) I believe that "not to follow through" usually means that you don't actually do what you said you would do.
(2) Let's say your childrem were naughty. You might say, "OK. No television for one week."
(a) But if you see that your children are very sad, then you might say,"OK. It's been only two days. But I will let you watch TV again starting today if you promise to be good next time." Of course, your children will agree.
(b) A week later, they are naughty again. You say, "OK. No dessert for one week." Your children will not be afraid, because they know that you never follow through on punishment. They know that your determination will weaken in one or two days and that soon they will again be enjoying cake and ice cream!!!