Cobordism invariance of the family index

We give a K-theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres over a compact base space B, and define a notion of cobordant families using K^1-groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push-forward map in K-theory to reduce it to families on B x R^n.