Does the tattoo stay the same size? Does it grow proportionally along with the baby? Does the ink become so diffuse do to growth that it essentially disappears? Or something else?
According to my wife the nurse, that tat of a butterfly on your little baby girl would end up a melted condor by the time she got to be an adult. The tat would still be there, but mis-shapen and stretched into who knows what.
Makes me think of The 30-Year-old Tattoo.
I’d encourage the baby to stick with pierced ears and hold off on tattoos for a few years.
You know babies – they’re so headstrong!
Especially avoid biker insignia.
Oh dear, I do hope the facial hair isn’t a tattoo.
Their heads are really mushy up tip.
Not until their fontanelle ossifies they’re not.
Agreed, IME. I’ve seen a lot of elderly (think saggy-skin type elders) and/or obese patients with greatly distorted tattoos gotten many years before. Reminds me of the joke “Why does a young woman get a tat of a small rose on her breast? To have a long-stem rose many years later!”
Some of the older tats I’ve seen have been hardly recognizable especially when person gains immense weight and/or garners more skin area than average person does, fwiw. I knew a set of twins (guys) who the parents had a small dot upon heel of one of them at birth - and when they were around 25-ish years old and each had huge feet relatively speaking (tall lanky guys), the 'dotted area was ~dime-sized and very faint; had to look hard to see it, to be honest.
A visit to Great Wolf Lodge in the winter is a pretty good living example of how well cough cough or not that tattoos age.
Nothin’ like a hawt young thing two years later with a couple of rug rats and 20 pounds (to be nice about it) of baggage, to showcase that the cutesy tweetie bird or whatever just doesn’t age well. Jus’ sayin’
"If you tattoo a baby, what’s the long term prognosis for the tattoo?"
I’ll bet (and hope) the short-term prognosis for the parents is a visit from CPS*!* 
As a nurse whose clientele is primarily the elderly, I have seen a lot of old tattoos, some of them 60 or 70 years old (that’s the tats, mind you, the OWNERS are in their 80s or 90s,) Most of the lines are blurred and diffuse to some extent, but from what I can tell most were not the high quality art that can be found today (not in every shop, but it can be found.) Also, people in the past did not protect their ink with sunscreen or have touchups like they do today. Weight gain and skin stretching can change the shape of designs as well and stretch marks can put gaps in the design.
Time does a number on our bodies as we age, by the time you are 80 or so, tattoos are usually the least of your worries. YMMV.
Aren’t most babies microchipped these days?
Seriously?! After which the parents are given tin-foil hats… 
The coloured ink is quite a lot more uv-stable now than it used to be (black was always stable). Which I think is kind of unfortunate, because I quite liked the idea of getting a tat that you could fade right out with simple uv exposure.
Thise that are set aside for the hunting preserves - yes.
Infants have a remarkable ability for self repair and regeneration even missing limbs can grow back. Perhaps possible that the tattoo would quickly disappear.
Given the tendency of toddlers to roam, microchips aren’t the worst idea in the world. Those baby leashes never look quite right.
Plus they can be monitored efficiently well into old age. I like it!