It depends if the mom is fed veggies and the babies are exposed to that from birth because the mom gets them. The mom helps pass the proper digestive bacteria to her kits to be able to tolerate eating veggiesand properly be albe to digestive them, while still babies. However, I wouldn't feed fruit or any other high carb foods(grains, carrots, etc) regardless, as baby rabbits digestive tract isn't developed enough to properly digest carbohydrates, and it can easily become overwhelmed with an excess of sugars and carbs, leading to serious enteric illness. But because this litter has been having issues and dying, I also would hold off on veggies now as well, even if they are used to getting them with mom, just in case the veggies and/or fruit could have lead to this happening.
The best thing for a rabbits digestive tract, is a medium coarse grass hay. Fresh grass that is soft early growth, can be too rich of a grass, especially for baby rabbits. They need a more mature growth with plenty of fiber in it, though still leafy enough for them to eat easily. It's the fiber in hay that is what helps regulate a rabbits digestive movement and microflora, which is what promotes good digestive health. If you don't currently feed a good quality grass hay, it's what I would do.
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I can't guarantee that nothing else will happen if you feed hay, or that this is the cause, as there just isn't enough in the way of symptoms beforehand, to help point to a more definite cause. Since mushy poop is the only thing you observed, and no other of your baby rabbits have gotten sick, that's my best guess as to what these kits may have possibly died from. The only way to know for sure would be to have a rabbit vet do a necropsy on one of the kits, and even then it can sometimes come out inconclusive.
The kits are best left with mom until at leas 7-8 weeks old, before getting weaned, if at all possible. The only reason I would wean any earlier is if she became aggresive with them, or if she is found to be the cause of her kits dying, due to passing on something like bacteria through nursing, or if she was found to have rabbit syphilis. Then I would wean immediately. So it would be a good idea to check your doe over very thoroughly, looking for any signs of mastitis, and checking her genitals and face for any sign of lesions or odd bumps.
If in doubt, and you have more rabbits dying, it would be best to consult with a knowledgeable rabbit vet, to try and find the cause. Espeically if you start having any adult rabbit fatalities.