I've had this problem in my previous life as a teacher.....
Are they asking you to do the DBS check yourself? Or just they need your info to carry out the check?
You, or the entity requiring the DBS, can ask the British Embassy or High Commission in the country you were residing in to request and authenticate a criminal record check from that countries authorities, that should then serve as the DBS. You'll need to provide them with the ID info in the country you're requesting the records from. Alternately you could go to the authorities where you are and request a criminal record check, or whatever their version of it is, but then you'd likely need to get it authenticated and apostilled for it to be accepted. If you go the route of getting the checks yourself, make sure you check with the relevant UK authorities what would be acceptable beforehand, as you can end up wasting time, effort and money! (I speak from experience!)
Also find out how far back they need the check to go, if you've lived in several countries you're going to have to do this process for each of them, unless they are only requiring records within a certain limit of time.
You're also likely to have to provide additional proofs of identity in order for them to request criminal records from overseas and to authenticate that the criminal record provided does indeed belong to you!
The UK government provides some guidance, but its a very boring read!
https://www.gov.uk/government/public...eas-applicants