So far as I am aware both Fidelity and Cofunds accept transfers into their wrap ISAs from other providers - this is called an "in specie" transfer facility meaning that the thing itself (e.g an existing holding of unit trusts in an ISA), can be transferred. However they won't permit transfers from their platform to other providers "in specie" insisting on conversion of the unit trusts to cash and then transferring the cash to the new platform/provider.
This acts as a significant disincentive to transfer because of the transaction costs and the uncertainty in value caused by the delay and being forced out of the market for some time - in effect a lottery in volatile markets. I believe that this is behaviour motivated wholly by the self interest of Fidelity and Cofunds. The fact that such transfers are entirely possible is evidenced by their acceptance of transfers to their platforms in this way.
Two or three years ago Citywire ran a brief campaign to get Fidelity and Cofunds to change but it quickly ran out of steam in the face of prevarication and a lot of smoke thrown down by Cofunds. Working on it, very complex, threeway transaction, working party etc, etc, - all very unconvincing.
This is the sort of thing that the FSA should pounce on because it is anti-competitive behaviour. Furthermore, this significant restriction isn't mentioned in any prominent way when people join these platforms.
Perhaps Citywire will relaunch its campaign given that the two or three years Cofunds said it needed to sort this out has now passed!! I don't think either Cofunds or Fidelity have changed their position nor are they likely to in the absence of real market pressure. Unfortunately not many folk pick up on this when looking for a home for their ISA savings.