Your question (as you asked it) is not about thermodynamic itself, but about kinetics. The simple short answer is yes. Thermodynamic process influences the phase transitions in different ways.
Thermodynamically, it you think about elementary thermodynamic process, that means that you fix some thermodynamic variables, such as T, V, the number of particles or chemical potential. Correspondingly, you need to consider the transition with these parameters fixed. Here it should be born in mind that during the 1st order transitions even in principle you cannot fix any pair of thermodynamic parameters. The evident example is say, the volume and the number of particles. The 1st order transition assumes the volume jump. If you somehow fix the volume, there is no volumetric jump, but the number of particles after the transition must change. In theory this means that for the description of a transition not each thermodynamic potential is equally good. For example, the free energy assumes exactly the two discussed variables, and strictly speaking is not a right potential. One should instead strictly speaking use the large omega-potential.
The second aspect is that the transition is characterized by its phase diagram. That is the plane of two (or space of more) thermodynamic parameters divided into regions of existence of various phases. For example, often one used the p-T plane as the plane of such parameters. On this plane the transitions will look as lines. When they meat with one another they make multiple points (typically - triple points).
From this perspective, the thermodynamic process is manifested as a line on this phase diagram along which our system is processed (moved). This line may be a straight line, say, p=0 (that is we only vary T), but it may be also a complex line. Depends upon your set up, if you are an experimentalist. Now, depending upon the shape of this line the system may pass through some of its phases and miss some others. It even may go through one and the same phase several times. Form this point of view the answer is also evidently, yes, it influences.
And now from the kinetic point of view the answer is also yes. You are right, that the heat may go out and hinder for some time propagation of a phase transition front, then the heat diffuses, the temperature goes down and the front again starts propagating, the heat goes out and the front stops and so on. Such effects are observed sometimes. May be also that the heat dissipates fast (as e.g. in metals), and the front moves smoothly. There are also a lot of other possibilities. I would say that this is still an open scientific area.