What if, instead of sending you into the past, weeping angels sent you into your future? Say the 'harvest' a 20 year old and send them into the future where they are 80 years old - the angel gets the energy of the 60 years.
The 20 year old would be sent 60 years into the future, but the 80 year old would not be there. When the 20 year old is sent to the future they "disappear" for the 60 years until they "reappear" once again.
In Doctor who the weeping angels send you back in time and feed off the energy of a life you could have lived.