I did not use "cry it out" but we followed other strategies that some relatives didn't. Frankly, it's impossible not to: between how long the child nurses for, cloth vs disposable, using a walker or not, how they go to sleep, how often they have a bath, what solid food you introduce first and so on it is simply a statistical impossibility that you did everything the same as every relative. Even for the "important" choices it is not possible.
So. My approach is NOT to convince these people that your way is right, or even that it is right for your children. I know this sounds counter-intuitive. After all, you're doing it because it's right, and they're interfering because they think it isn't right, so if you just fix their misconception, they'll stop interfering, right?
But no. They are interfering because they think their opinion is relevant. It is actually easier to convince even very close relatives that their opinion is not relevant than to get them to change that opinion. It's more pleasant, too. Instead of barraging them with arguments about why they are wrong (and possibly even were wrong when raising one of you) you simply remind them that you are the parents and this is what you are doing. You thank them for their advice if advice is being offered. You reassure them of the baby's health and happiness if they seem to be worried about that. And you stick firmly to the position of "this is what we have decided to do in our family."
This worked wonders with a relative who felt that four months was the longest a baby should get any breastmilk at all and was seriously perturbed that weaning wasn't even started yet, much less completed, as each of my children reached that age. We didn't tell her any facts, studies, or doctor opinions. We didn't rebut her facts and studies and decades earlier doctor opinions. We just said things like "I know" and "I remember you telling me that" and just kept right on raising our own children. When asked "aren't you going to wean soon?" we said either "no" or something similarly simple, including "we'll let you know!" When prompted to do something at bedtime that wasn't a bedtime thing for us, we said "we've tried that, and this is what we do now." Had someone offered (or flat out just tried to go and do) to do something that wasn't part of our routine, we would have stopped them in the same polite way you stop relatives from cleaning your bathtub or other too-intrusive helping. "Mom, please, no! Really! I can't let you do that!" We didn't include any right-and-wrong arguments (in your case "it will take days for them to get the routine back if you disrupt it" or the like) because (and this is the key) she didn't get a vote so she didn't need to be convinced.
You may feel the first time you play the "we're the parents" card without using any facts to back it up that you will crumble beneath the knowledge and experience of Aunt Alice or your mother or your father in law. But my experience (and that of many of my friends) is that you get your way and the arguments stop. There's nothing to argue with. You're not saying your way is better or right. You're just doing it your way and that's that. You're the parents. Try it.