The Linga Sarira or Subtle Body guarantees the continuance of identity of karma. Shankara explains:
....to the soul remaining still surrounded by the subtle elements, occur such thoughts about the future body as are called up by the accumulated results of past actions; and this expectancy becomes lengthened out to the next body like a leech. This being the manner of acquiring a fresh body, as shown by the Upanishads all other theories arising from the human intellect, such for instance as (the Sankhya theory) that when the all-pervasive senses and soul acquire a new body as a result of past actions, they start functioning there itself; or (the Buddhist theory) that the soul alone, by itself, acquires its function there, while the senses, just as much as the body, are born afresh in those different spheres of experience; or (the Vaisesika view) that the mind alone proceeds to the new place of experience; or (the Jaina view) that the soul alone jumps from one body to another like a parrot from one tree to another – all these are to be ignored as running counter to the Vedic view.
What your concept of personal identity is in this life will condition the view of your post-mortem identity. If you believe in the continuance of the karmic adventures of the Jiva or individual person after the 'great change' as James Carlyle, Thomas's father called it, then there must be continuity. The jiva in Vedanta is a body/mind entity so the subtle body must form a bridgehead to the next life. The material support for life is so to speak subtilised and this carries through the karma to its next venue.
Shankara indicates in a curtailed form how the different concepts of identity in this life, Sankhya, Jain, Vaisesika and Buddhist are reflected in the accounts of transition to the next. All show a definite metaphysical consistency and differ from each other.