  In this paper we construct time-dependent solutions of three-dimensional
gravity in AdS space dual to systems with boundaries (BCFTs), following the
AdS/BCFT prescription. Such solutions can be discussed in the context of the
dynamics of first order phase transitions, or more generally, in the
description of quantum quenches. As an example, we apply the holographic model
to calculate the dynamics of the entanglement entropy of a local quench
corresponding to a nucleation of a Euclidean bubble. As in the known 1+1 CFT
examples of local cut and glue quenches, the holographic entanglement entropy
grows logarithmically with time with the correct universal coefficient.
However, in the bubble quench, the behavior is different at late times. The
AdS/BCFT model exhibits the light-cone spreading of correlations and saturation
at late times. We also find an analytical formula for the entropy at finite
temperature. In the latter case the initial logarithmic growth is followed by
the linear law at intermediate times.
