A responsible network operator will require you to prove that you are a network operator (this may be subject to government licensing) or prove that you own all the number ranges that you intend to include in your caller ID values. This means that you will not be able to forward an arbitrary incoming caller ID.
Alternatively, they may set the unscreened attribute on the caller ID. In that case, if the call leaves part of the network that supports that attribute, the caller ID should be suppressed, at that point. An example of this would be a callee using an analogue phone. (I’m not sure whether GSM phones can indicate unscreened status.)
If you are lucky, they may set screened for ranges that you have registered with them and use unscreened for numbers for which they do not have proof that you own them.
The number of bogus caller IDs reaching the UK from scammers and less reputable telesales companies, using VoIP, suggests that there are quite a few irresponsible network operators.