Integral relations based on boundary layer theory are derived to study the motion of an isolated, two-dimensional thermal plume through a viscous mantle containing polymorphic phase changes. Analytical results are obtained which show that phase transitions alter average mangle convective velocities by less than 50%. In particular we find that the olivine-spinel transition, approximated as univariant, can enhance the circulation velocity of upper mantle convection by 30--40%, while it can enhance the overall amplitude of whole mantle convection by a few percent only. Our calculations demonstrate that a possible endothermic phase change located at 650 km will not prevent deep mantle convection by 30--40%, while it can enhance the overll amplitude of whole mantle convection by a few percent only. Our calculations demonstrate that a possible endothermic phase change located at 650 km will not prevent deep mantle convection unless the Clapeyron slope defining the transition exceeds -0.3 kbar/¿K. This large value is more than one order of magnitude greater than what has been proposed for the 650-km discontinuity. We then extend the method to include compositional buoyancy and effects of the divariant nature of the olivine-spinel transition. Analysis of the motion of a compositionally buoyant plume (one having an anomalous Mg/Fe ratio relative to the ambient mantle) reveals that the chemical plume locally distorts the transition in a way which contributes buoyancy and enhances convective amplitudes by 10% or less. Finally, we combine thermal and compositional buoyancy to investigate the interaction between a thermochemical plume and a compositionally induced density interface. The results are used to determine the stability criterion for two-layered mantle convection. We find that a compositionally produced density increase of 4% at 650 km is needed to prevent mixing between the upper and lower mantle. Seismological observations indicate that density jump at the 650-km discontinuity is between 5% and 9%. If all the density increase observed there is due to change in composition, then two-layered mantle convection is stable. However, if phase transitions account for part of the density increase, so that the chemically produced component is less than 4%, then two-layered mantle convection may be unstable.