Here is Mr. Hingston's advice, as supplied to me, complete and unadulterated.
Please note: The following advice is relevant only in Sotland.
"The answers are to be found partly in the terms of the Deer (Scotland) Act 1996 and at common law.
At common law, deer are wild animals [unless they are being farmed] and thus belong to no-body. That is why the offence of poaching had to be made by statute, which in Scotland is the '96 Act. The offences are created by the sections in Part 111 being sections 17-24. Exceptions to these rules are found in sections 25 and 26.
It is an offence for someone, without legal right or without permission from someone having legal right, to kill or injure a deer [s17(1)] or to remove a deer carcase [s17(2)] on or from any land. However these offences are subject to the exception in s25, where it is not an offence against s17 if the act is to prevent suffering to inter alia an injured deer.
Hence the answer to your question is that it is not an offence to follow a wounded deer from land where you have permission to shoot onto other land and to kill it on the other land if that killing was was for the purpose of preventing suffering to it. I would have thought it is incumbent upon any sportsman, concerned as he should be with the wellfare of his quarry to make sure it was not suffering, to take steps to put it out of its misery.
Because it is a wild animal it does not belong to the owner on whose land it was first shot nor to the owner upon whose land it died. Ownership of the ground conveys no ownership rights to the animal whether dead or alive. At common law it belongs to the person who kills or takes it which explains why the court should order forfeiture otherwise it would have to be handed back to the poacher! The offence of removing the carcase [s17(2)] does not apply because it was killed in terms of s25.
That is the legal position, proper manners and law are however different. It would be a matter of courtesy [and may avoid false allegations of poaching] to tell the relevant landowner what happened and explain why you were shooting on his land.
David Hingston"