Concerning your first question, here is an example from (microlocal) analysis and symplectic geometry: Fourier Integral Operators (FIO) correspond to canonical relations which are Lagrangian submanifolds of $T^*Y\times T^*X$, where $X$ and $Y$ are the source and the target manifold, respectively. Composition of FIOs corresponds to the composition, as relations, of their underlying canonical relations. The solution operator of (the Cauchy problem of) the wave equation is an FIO; here two points in the cotangent bundle are related if they are on the same bicharacteristic (=solution curve of Hamilton's canonical equations). The pull-back by a smooth map $f:Y\to X$ is an FIO, $f^*$. Here the relation is the conormal bundle of the graph of $f$ (after one of the fiber variables has been multipled by $(-1)$). In case $f$ is a diffeomorphism, the associated symplectomorphism $T^*Y\to T^*X$ is the canonical relation of $f^*$. The FIOs which correspond to the identity on $T^*X$ are precisely the pseudo-differential operators on $X$. When dealing, for example, with wave propagation in manifolds with boundaries, the FIO calculus, of which an essential part is the composition of canonical relations, is very convenient and useful.

Concerning your first question, here is an example from (microlocal) analysis and symplectic geometry: Fourier Integral Operators (FIO) correspond to canonical relations which are Lagrangian submanifolds of $T^\ast Y\times T^\ast X$, where $X$ and $Y$ are the source and the target manifold, respectively. Composition of FIOs corresponds to the composition, as relations, of their underlying canonical relations. The solution operator of (the Cauchy problem of) the wave equation is an FIO; here two points in the cotangent bundle are related if they are on the same bicharacteristic (=solution curve of Hamilton's canonical equations). The pull-back by a smooth map $f:Y\to X$ is an FIO, $f^\ast$. Here the relation is the conormal bundle of the graph of $f$ (after one of the fiber variables has been multipled by $(-1)$). In case $f$ is a diffeomorphism, the associated symplectomorphism $T^\ast Y\to T^\ast X$ is the canonical relation of $f^\ast$. The FIOs which correspond to the identity on $T^\ast X$ are precisely the pseudo-differential operators on $X$. When dealing, for example, with wave propagation in manifolds with boundaries, the FIO calculus, of which an essential part is the composition of canonical relations, is very convenient and useful.

Concerning your first question, here is an example from (microlocal) analysis and symplectic geometry: Fourier Integral Operators (FIO) correspond to canonical relations which are Lagrangian submanifolds of $T^*Y\times T^*X$, where $X$ and $Y$ are the source and the target manifold, respectively. Composition of FIOs corresponds to the composition, as relations, of their underlying canonical relations. The solution operator of (the Cauchy problem of) the wave equation is an FIO; here two points in the cotangent bundle are related if they are on the same bicharacteristic (=solution curve of Hamilton's canonical equations). The pull-back by a smooth map $f:Y\to X$ is an FIO, $f^*$. Here the relation is the conormal bundle of the graph of $f$ (after one of the fiber variables has been multipled by $(-1)$). In case $f$ is a diffeomorphism, the associated symplectomorphism $T^*Y\to T^*X$ is the canonical relation of $f^*$. The FIOs which correspond to the identity on $T^*X$ are precisely the pseudo-differential operators on $X$. When dealing, for example, with wave propagation in manifolds with boundaries, the FIO calculus, of which an essential part is the composition of canonical relations, is very convenient and useful.

Concerning your first question, here is an example from (microlocal) analysis and symplectic geometry: Fourier Integral Operators (FIO) correspond to canonical relations which are Lagrangian submanifolds of $T^*Y\times T^*X$, where $X$ and $Y$ are the source and the target manifold, respectively. Composition of FIOs corresponds to the composition, as relations, of their underlying canonical relations. The solution operator of (the Cauchy problem of) the wave equation is an FIO; here two points in the cotangent bundle are related if they are on the same bicharacteristic (=solution curve of Hamilton's canonical equations). The pull-back by a smooth map $f:Y\to X$ is an FIO, $f^*$. Here the relation is the conormal bundle of the graph of $f$ (after one of the fiber variables has been multipled by $(-1)$). In case $f$ is a diffeomorphism, the associated symplectomorphism $T^*Y\to T^*X$ is the canonical relation of $f^*$. The FIOs which correspond to the identity on $T^*X$ are precisely the pseudo-differential operators on $X$. When dealing, for example, with wave propagation in manifolds with boundaries, the FIO calculus, of which an essential part is the composition of canonical relations, is very convenient and useful.