@goplacidlyamidthenoise
I have done exactly this with some shares held in my Equiniti account. Equiniti are the Registar, so manage them for you.
Given that I didn’t know the exact cost when I obtained them (most came from my mum when she died, parents had got them in the 1980s share sell off), I have just transferred £3k worth per tax year, so I know that I am within my CGT allowance.
Assuming that your child/ren are over 18, it is very easy to do, there is a transfer form provided by Equiniti.
I will see if there is a link to it online, but if not, I’m happy to send you the blank version via a private message. You need to fill one in for each ‘child’ but can then send the form(s) in the same envelope along with share certificates totalling the number of shares being transferred.
I always send them registered, DC usually get their share certificate back within 2 weeks, they can then set up their own online Equiniti account if they don’t already have one.
Equiniti are very good at assisting, I have emailed them a few times about things, no charge for any advice.
Oh, and before starting my transfer over last couple of years, I sold one set of shares, and that cost me approx £50. Much cheaper (free) to transfer them!