The RBaCo4O7 (R = Ca, Y, Tb, Ho, Tm, Yb, Lu) compounds provide a novel topology for studying the competition between triangular geometry and magnetic order. Here, we report the structural and magnetic behavior of the Lu member of this series via neutron and synchrotron x-ray diffraction, magnetization, and resistivity measurements. We determined sequential phase transitions and a strong competition between a stable and a metastable low-temperature state that critically depends on controlled cooling rates and the associated heat removal kinetics. No evidence for long-range ordered magnetism was detected by neutron diffraction at any temperature. However, very slow spin dynamics are evidenced by time-dependent neutron diffraction measurements and can be explained by several competing magnetic phases with incommensurate short-range correlations coexisting in this material.