Layer, layer, layer
For a baby under 6 months old that’s still on your front, they probably only really need one or two extra layers compared to you. I’d also only use wool, as fleece and other non-natural fibres can’t breathe, so the baby can overheat as they can’t regulate their body temperature well yet. If it’s windy or raining, just put your raincoat over both of you – a larger size raincoat should be fine for a smaller baby, or invest in a babywearing soft-shell jacket. Because the baby is right up on your chest, you’ll be able to feel how warm they are at all times. Just make sure their breathing area is clear of any obstructions, and don’t forget to keep an eye on fingers and toes.
For kids older than 6 months, make sure you have a raincover for your hiking pack. And even though I’m a fan of natural fibres, fleece is definitely superior in keeping you warmer on the trails. I tend to use wool thermals on the bottom, with polyprop & fleece on top. And if it’s windy or raining, a raincoat and a pair of waterproof pants.
For kids younger than about 2.5, gumboots are the perfect footwear, as they’re 100% waterproof, and your under 2.5 year old is unlikely to be walking large distances yet, so comfort doesn’t matter too much yet. Just make sure they’ve got thick wool socks, or two pairs of thin ones. Fleece gumboot liners are also a good idea.
We also love our daughters fleece onesie. It means no cold hips or backs, and so is great not only for sleep, but for hiking in also.