The most natural phrase is: The shoplifter died (while) in police custody. The shoplifter died (while) in police detention. There may well be some technical/legal difference between the two, which would vary from one country/state to another. Restraint is isn't - as far as I know - used in this sense. Restraint suggest a more physical restriction eg being put in handcuffs.

Here is how I would write these sentences... The shoplifter died WHILE BEING DETAINED BY POLICE. The shoplifter died WHILE IN police custody. The shoplifter died during police restraint. This sentence is not good because it suggests that the shoplifter died while the police were restraining him or her (while they were arresting them... not while they were in jail). If this is not your intended meaning then I would not use this sentence. Hope this helps, Nick

They are all correct, but are not how they would all be commonly phrased mainly because of the use of during. There are many ways these statements could be said, but below is how I as a native speaker would feel most natural saying these phrases. 1) 'The shoplifter died during detention by police' would be better phrased as 'The shoplifter died while in police detention.' 2) 'The shoplifter died during restraint by police' would be better phrased as 'The shoplifter died while being restrained by the police. 3) 'The shoplifter died during police custody' would be better phrased 'The shoplifter died while in police custody.' Hope this helps!

@Warren: Many Thanks, it makes sense:)

during means over a period of time, so during the last war many children starved to death. In your examples the time ends with the death of the shopkeeper and during is used for the period of police action so it doesn't work very well. Better might be - the shopkeeper died while in police detention/custody. the shopkeeper died while being restrained by the police. during police custody he was taken to a cell and beaten with a knout.-here something happens during the period of time.