A proper conformal Killing vector (CKV) in a fluid spacetime will be defined to be inheriting if fluid flow lines are mapped conformally by the CKV. The consequences of this definition are considered. In particular, a general class of spacetimes called synchronous spacetimes are investigated and it is proved that orthogonal synchronous perfect fluid spacetimes, other than Friedmann-Robertson-Walker spacetimes, admit no proper inheriting CKV. Generalizations of this result to non-comoving perfect fluid and comoving but non-perfect fluid synchronous spacetimes are then considered. Proper CKV spacetimes, and especially inheriting CKV spacetimes, are very rare, and a determination of all such spacetimes is of interest. In particular, it is conjectured that a non-existence result of the above form may be valid when generalized to spacetimes other than synchronous spacetimes (at least in the perfect fluid case).