It sounds like you are groping your way to a transmission line model understanding of what's happening.
With models, we like to use the simplest model that captures the performance of the circuit we're working on. For most people, for most of the time, that's the circuit theory model. In it, wires have zero resistance, constant voltage everywhere, they transmit effects instantaneously, and electrons don't exist, just current and voltage.
However, you are worrying about time, so it's time to take one step up, and consider the transmission line model.
Take a long line, connected at the far end to a resistor to ground. There's a fast DMM at both ends of the line. At t=0, you connect a voltage source with a certain output impedance to the line.
The input end of the line jumps up to some voltage. A voltage wave sets off along the line, and a current wave with it, the current wave charging the capacitance of the line up as it goes. It's not a result of the voltage wave, it doesn't cause the voltage wave, but the two go hand-in-hand along the line. Their ratio depends on the geometry of the line, the square root of the ratio of the line's inductance and capacitance, and is called the line characteristic impedance. This has nothing to do with the line's resistance, which can be assumed to be zero without affecting this first order description of what happens.
The initial voltage that the near end jumped up to was set by the ratio of the source output impedance and the line impedance. It has nothing to do with the resistor at the far end. At t=0, the far DMM is still reading 0 volts. The source end of the line doesn't 'know' anything about the far end.
After the wave has propagated down the line at the speed of light in the line (often around 0.6c for plastic insulated lines), it reaches the resistor. The far DMM jumps up to some voltage. The voltage it jumps to is determined by the line impedance, and the value of the far resistor. If these are not equal, a return wave is generated, and this sets off back down the line again, until it reaches the source. A further reflection may or may not happen, depending on the value of the source impedance and the line impedance.
Each time a reflection is generated, it's smaller than the previous one, and the wave will have a new voltage and current amplitude. After sufficient number of reflections, the line settles down, and the source and resistor have at last 'agreed on' a voltage and current that suits them both. This means that the voltage and current are only determined by the source impedance and the load resistance, and not the characteristic impedance of the line.
For short lines and slow instruments, you can ignore all this detail, and just use circuit theory as a quick, and perfectly adequate, way of figuring out what's going to happen when you connect a battery to a bulb, or a transistor to a biassing circuit.
I've not mentioned electrons in this newer model. They don't really help, not with understanding anyway. If you must, consider them as the mechanism that underlies charge in metal conductors (other charge carriers exist in other conductors, don't get chauvanistic about electrons). Positive charge is a local deficit of them, positive charge a local buildup, and current the net rate they move from one place to another.
If you really, really, want to worry about electrons, look up the Drude model. Be warned that it's not very accurate, has little predictive power, and is a hand-waving classical approximation to what really needs to be a quantum mechanical description, but some people find it helps. Working electronic engineers can safely ignore electrons, they are more the domain of physicists.
