  We formulate dynamical phase transitions in subsystems embedded in larger
quantum systems. Introducing the entanglement echo as an overlap of the initial
and instantaneous entanglement ground states, we show its analytic structure
after a quench provides natural definition of dynamical phase transitions in
the subsystem. These transitions come in two varieties, the entanglement-type
transitions and the bulk-type Loschmidt transitions. The entanglement-type
transitions arise from periodic reorganization of quantum correlations between
the subsystem and its environment, manifesting in instantaneous entanglement
ground state degeneracies. Furthermore, the entanglement echo distinguishes the
direction of the quench, resolves spatially distinct dynamical phase
transitions for non-uniform quenches and give rise to sharply-defined
transitions for mixed initial states. We propose an experimental probe to
identify entanglement-type transitions through temporal changes in subsystem
fluctuations.
