Suppose we have a family of compact oriented even dimensional spin manifolds $\{Y_x\}$ parameterized by a compact even dimensional manifold $X$. The $Y_x$'s are all diffeomorphic to some $Y$, of dimension $n$, and fit together to form a fiber bundle $\pi : Z \rightarrow X$ with fiber $Y_x=\pi ^{-1}(x)$. $TZ$ has the subbundle $V:=\text{ker }\pi_*$ which is tangent to the fibers. There may be a family of coefficient bundles also and we obtain a family of twisted Dirac operators $D_x:\Gamma(S^+_x\otimes E_x)\rightarrow \Gamma (S^-_x\otimes E_x)$. The index of the family gives rise to an element $\text{ind} D \in K(X)$, which is the virtual vector bundle $[\text{ker } D_x]-[\text{coker }D_x]$ when the dimension of both spaces are constant. Finally, there is a map $\text{H} ^{\*}(Z,\mathbb{R})\rightarrow \text{H} ^{\*-n}(X,\mathbb{R})$ known as the Gysin homomorphism or integration over the fibers map. We'll use the latter terminology writing the map $\int_Y$ and regarding cohomology classes as living in de Rham cohomology. The Atiyah-Singer index theorem gives

$$\text{ch }(\text{ind } D)= \int _Y \hat A (V) \text{ch}(E)$$

What general results exist regarding the components of the Chern character of the index bundle, or equivalently the results of the integration over the fibers map, for twisted Dirac operators?

To illustrate, an immediate answer is that the zero cohomology (virtual rank) is the index of the Dirac operator on $Y$. A more interesting answer is that in some cases that might be all one obtains: it is a result of Borel-Hirzebruch that the signature is strictly multiplicative in all bundles where $\pi_1$ of the base acts trivially on the rational cohomology of the fibers. The signature is the index of a certain twisted Dirac operator. If we have a family of these operators such that $Z\rightarrow X$ satisfies the condition involving the fundamental group, then the strict multiplicativity gives $\text{ch}(\text{ind }D)=\int_Y \hat A (V)ch(E)=\text{sign }(Y)$. A priori one could expect higher degree cohomology classes. It seems interesting that these vanish.

If the question is too vague or broad, I would be happy knowing

Are there any instances in which there are known relations between the Chern character of the index bundle and the Chern classes of $X$?

Suppose we have a family of compact oriented even dimensional spin manifolds $\{Y_x\}$ parameterized by a compact even dimensional manifold $X$. The $Y_x$'s are all diffeomorphic to some $Y$, of dimension $n$, and fit together to form a fiber bundle $\pi : Z \rightarrow X$ with fiber $Y_x=\pi ^{-1}(x)$. $TZ$ has the subbundle $V:=\text{ker }\pi_*$ which is tangent to the fibers. There may be a family of coefficient bundles also and we obtain a family of twisted Dirac operators $D_x:\Gamma(S^+_x\otimes E_x)\rightarrow \Gamma (S^-_x\otimes E_x)$. The index of the family gives rise to an element $\text{ind} D \in K(X)$, which is the virtual vector bundle $[\text{ker } D_x]-[\text{coker }D_x]$ when the dimension of both spaces are constant. Finally, there is a map $\text{H} ^{*}(Z,\mathbb{R})\rightarrow \text{H} ^{*-n}(X,\mathbb{R})$ known as the Gysin homomorphism or integration over the fibers map. We'll use the latter terminology writing the map $\int_Y$ and regarding cohomology classes as living in de Rham cohomology. The Atiyah-Singer index theorem gives

$$\text{ch }(\text{ind } D)= \int _Y \hat A (V) \text{ch}(E)$$

What general results exist regarding the components of the Chern character of the index bundle, or equivalently the results of the integration over the fibers map, for twisted Dirac operators?

To illustrate, an immediate answer is that the zero cohomology (virtual rank) is the index of the Dirac operator on $Y$. A more interesting answer is that in some cases that might be all one obtains: it is a result of Borel-Hirzebruch that the signature is strictly multiplicative in all bundles where $\pi_1$ of the base acts trivially on the rational cohomology of the fibers. The signature is the index of a certain twisted Dirac operator. If we have a family of these operators such that $Z\rightarrow X$ satisfies the condition involving the fundamental group, then the strict multiplicativity gives $\text{ch}(\text{ind }D)=\int_Y \hat A (V)ch(E)=\text{sign }(Y)$. A priori one could expect higher degree cohomology classes. It seems interesting that these vanish.

If the question is too vague or broad, I would be happy knowing

Are there any instances in which there are known relations between the Chern character of the index bundle and the Chern classes of $X$?