Before you think about fire lapping evaluate the gun. Clean that new barrel very good before the first round. Then I use the shoot and clean after each round routine. While you do this, note how much copper fouling you have. If it isn't real bad, and you don't have to spend hours getting the copper out between rounds I wouldn't think about lapping. Just do one of the shoot and clean routines.
I've Fire Lapped two rifles, both would copper foul so bad that I had to spend two days cleaning between each shot and still not get all the copper out. The first rifle was a .338 Win Mag. and the second was a .243. After doing the fire lapping to the .338 which took about a week for me to do, I cleaned it real good and went to the range, shot two rounds to season the barrel. Then I shot my first group and I was pretty happy with myself for being such a LUCKY shot, as all 5 holes were touching. Then I shot another, and another. I shot some groups that looked like a .338 bullet lying on its side! Da*m tight! I was happier than that proverbial pig. But it still copper fouls. Not as bad, but still requires allot of cleaning. But I can't complain because of what it did for the accuracy.
Then I decided to make that .243 shoot one hole 5 shot groups also. I did the fire lapping on it. Helped the fouling some, didn't improve the accuracy that much. And now after doing the lapping the throat is soooo long that I can't get close to the lands with any bullet, even a 100 gr bullet seated way out in the end of the case doesn't touch them. Now this gun still shoots pretty darn good. Sub moa, sometimes I get al 5 shot holes touching. But I know that I severely shortened this barrel's life by doing the lapping. And I was pretty conservative with the lapping procedure.
So.... Why do you want to do it? What do you want to gain? Can you put up with a little more cleaning than you like? What cal. is the rifle? What do use it for? How often will you need to clean it? All questions you need to answer for yourself. Then decide which route you want to go. All lapping is wear on the barrel. I think that if you have a problem in the bore, i.e. rough spot, then hand lapping will be of more help because you can concentrate on that spot. If it's a fouler, and you can't keep up with the cleaning, then go ahead and fire lap her. But I suggest you start conservatively, watch that throat, especially if your talking about the small calibers.