In order to move electrons in a material, we can apply an external electric field. In this case electrons (negatively charged) will flow towards the positive terminal. The resulting flow is what defines an electric current. Thus, we are not applying a current, but actually applying a voltage, which results in current. As for the path the electron takes, if we are talking in macroscopic terms, the electrons will take the "least resistive path," where resistance is a function of both the material and path length. If you are considering more fundamentally, how electrons move through a particular material, then this becomes somewhat more complicated, and dependent on the material. In good conductors, such as metals, electrons are weakly bound to the individual atoms that make up a lattice, and are "free" meaning they can easily migrate throughout a material (even without an applied field.) In semiconductors, such as silicon, there is a band gap, which means there needs to be a driving force (electric field) of sufficient energy to begin extracting any current at all, and these currents can either be caused by injecting electrons or the opposite, collecting positive charges or "holes." In organic semiconductors and other highly disordered materials, there is a less well defined lattice system, and charges are carried through a "hopping mechanism." This means charges are transferred from molecule to molecule as it moves across the material. In each case there are well defined theories of charge carriers mechanics and the mechanisms can change when you are dealing with different magnitudes of voltages or different temperatures. I know this is an incredibly broad answer, so if you have a more specific question, or specific material system/device set-up, I would be happy to try to help more.

At this point, we are probably getting out of my expertise, but I would say in general the electrons will not flow in a straight line. At the molecular level, in a fluid, there is constant motion. The conductive pathways in a fluid are far more complicated and rely on molecular motion. Typically it is ions, or solvated molecules which actually conduct the charges. Temperature will increase this motion, and thus likelihood/rate of conduction, but I doubt it would change the physical path of electrons too much. Applying a higher voltage should create a larger driving force for conduction, which would likely cause a more direct path, however I do not know how one would probe this experimentally.

Though it may not relate, 4 years ago two scientists Humphrey Maris and Wei Guo were able to photograph the motion of electrons in liquid helium. (The article is in nature, but lots of articles have been written about it). They say they seem to move in relatively straight lines, but that is over much more macroscale dimensions.