Experts have warned, however, that "clean eating" trends, such as going gluten or dairy-free - with no medical reason - could lead to nutritional deficiencies in children and set up unhealthy relationships with food.
In June, researchers in France said non-dairy milks were dangerous for babies, in a study of parents who had chosen to replace breast or formula milk with an alternative before their child had reached the age of 1.
A third of the children were found to have malnutrition. And in recent years there have also been reports of children raised on a vegan diet being admitted to hospital suffering with malnutrition.
Meal times with a toddler are enough of a battle for most parents so how responsible, or even practical, is it to raise one on such a restrictive regimen?
Karris McCulloch has been vegetarian for most of her adult life, and vegan for six years, and she and her husband were in no doubt that they wanted their children also to follow a wholly plant-based diet, with no meat, fish or animal products.
"There's a lot of scaremongering about it, and we're always told that vegans need to be especially careful, but I find that vegans are actually more clued-up about nutrition than most," says Karris, 32, who lives with her husband, Scott and two children Casey, 6, and Tyler, 2, in Glasgow where they run a food subscription service, The Vegan Kind.
"When my children were born, I was referred to a nutritionist because we were vegan, but when I went to the appointment I knew more than she did.
"I know how important it is to get protein, calcium, iron and vitamin B12. You can get these plentifully in plant-based foods - so we have lots of stir fries, with tofu, which is excellent for calcium and protein, or spaghetti bolognese with a soya mince, and they have soya milk that's fortified with calcium and things like avocado and nut butters for healthy fats.
"But we also give a multivitamin supplement, too, just to make sure.
"I think if you compared their diets to an average 6 or 6-year-old, they'd probably be healthier."
Lizzy Tighe, an artist and mother of four from Herefordshire, raised her children, aged 14, 12, 10 and 5, on a largely vegetarian diet and the family went fully vegan 18 months ago.
"My children suffer from eczema and asthma and it really helped - you could see a change in their skin," says Lizzy who blogs about healthy vegan living at Forking Fit.
She also gives her children fortified cereals and milk, and supplements.
"You do need to do your research, especially with young children, and be prepared to be inventive - you don't want to be a junk food vegan."
Growing children have unique dietary requirements for linear growth, weight gain and the development of the brain - and it's now established that nutrition in the first two years of life is vital for growth potential and programming for disease in later life.
Experts say it's possible, with advice and planning, to raise children safely on vegetarian and even vegan diets, and increased interest means there's now a wealth of information available on how to do it.
"Until fairly recently, we tended to advise against vegan diets for babies," says Mary Fewtrell, a professor of paediatric nutrition at University College London.
"It can be done, but just not on a whim. Children need supplements, for example for vitamin B12, as they won't get their full requirement from food."
Another concern is simply energy, she says. "Vegetables contain a lot of fibre so if a child is filling up on these, they might not end up consuming enough high calorie, high protein foods."
Of greater concern, however, is that the fashion for so-called clean eating - restricting or eliminating foods such as gluten and dairy - is being applied by parents to children.
The demonisation of these food groups, fuelled by claims by celebrities and bloggers, has seen soaring numbers of people cutting them out, often because of a perceived intolerance or allergy.
This is despite the fact that, beyond coeliac disease - an immune response to gluten, affecting around 1 per cent of us - the notion of gluten "intolerance" or "sensitivity" has been debunked.
More than half of British people had recently bought a "free from" product, according to a survey in May. Now, even baby and children's foods are being marketed as gluten or dairy free.
"We're certainly seeing adult trends in nutrition and food exclusion starting to translate to children," says Lucy Upton, a paediatric dietitian at Birmingham Women's and Children's NHS Foundation Trust and spokesman for the British Dietetic Association. "If you're cutting out something like gluten, you're cutting out a huge number of foods, which is going to make life hard."
The growing interest in exclusion diets among parents seems related to heightened concerns about allergy among the middle classes.
A third of British parents now believe their child has a food allergy, according to a 2015 study by allergy specialists - yet only one in 20 children would pass a clinical diagnosis. The researchers found allergies were much more likely be reported by parents in affluent areas.
The problem is confused by the fact that allergy does seem to be genuinely increasing in children, says Upton.
"Were seeing a rise in true allergy and a rise in parental anxiety - which corresponds with adults diagnosing themselves with things like gluten intolerance. But we would never recommend excluding anything without good medical reasons."
Attracting particular controversy is cow's milk protein allergy, which is notoriously difficult to diagnose. It affects up to 4 per cent of children, and some cases are unequivocal - where the child suffers a dramatic reaction with hives, swelling and vomiting, soon after consuming milk.
However, some children have a different form known as non-IgE allergy, where the symptoms - patches of eczema, diarrhoea, reflux, colic - are more vague and overlap with things seen commonly in healthy babies.
Parenting forums are awash with exhausted parents who report their child's symptoms improved dramatically as soon as they swapped their milk for a dairy-free alternative.
But while many parents have a long and frustrating wait before getting a proper diagnosis, others are mistakenly self-diagnosing an allergy or even have had a doctor misdiagnose it in their child.
"There's a paradox where we're not picking up the right children, but also inappropriately cutting out milk in the wrong children," says Upton.
British NHS statistics show prescriptions for formula to treat babies with cow's milk allergy rose by almost 500 per cent in the 10 years to 2016.
"Part of the problem is parents seeking information online, she says, which varies in quality.
"There are all these nutrition noises in the back of your head, that you might have read online or heard at the antenatal group. Lots of these symptoms can be quite normal, but people are not getting the right advice and are self-diagnosing."
She says parents should seek medical advice before cutting a major food group out of their child's diet - and to diagnose allergy, a doctor should take a thorough medical history including any history of allergy in the family.
"If you're not sure, you can try putting them on a milk-free diet for four weeks and see if things get better or not."
If a child is under 1, and has a true cow's milk allergy, they should have a fortified dairy-free formula milk; breastfeeding mothers may also need to remove dairy from their diet.
Unfortified nut milks and oat milks are not recommended for babies, as they are too low in calories, and are missing other important nutrients including iodine and calcium.
Aside from the effect on children's nutrition, experts say the unnecessary removal of certain foods from children's diets can cause youngsters to become anxious about them.
"Eating should be enjoyable and sociable, and anxiety is very infectious," says Upton.
"If they're having things whipped out of their hands, what will that do to their wider relationship with food?
"In a way it can make [children] feel there's something wrong with them," adds Fewtrell.
"It's hard enough for parents where children actually do have an allergy, so you want to make sure what you're doing is necessary."
Vegan mother Karris McCulloch takes this issue seriously.
"I think it's important to distinguish between fads and ethical food choices - a lot of people have jumped on the gluten-free bandwagon because they think it's healthy or helps you lose weight. That's passing on unhealthy lifestyle choices.
"Casey is fully aware of the reason we don't eat meat and dairy - we've simplified it and say that we love animals and we don't need to eat them to be healthy.
"We've also tried to teach her that nothing is off limits - so if we're at a party and she's offered a sausage roll, I tell her she is free to try it if she wants to.
"We want her to feel that she is choosing - forcing anything on a child isn't right."
