  The paper studies an Allen-Cahn-type equation defined on a time-dependent
surface as a model of phase separation with order-disorder transition in a thin
material layer. By a formal inner-outer expansion, it is shown that the
limiting behavior of the solution is a geodesic mean curvature type flow in
reference coordinates. A geometrically unfitted finite element method, known as
a trace FEM, is considered for the numerical solution of the equation. The
paper provides full stability analysis and convergence analysis that accounts
for interpolation errors and an approximate recovery of the geometry.
