Gene's point about safety is most important so let's always keep that out front in any discussion.
In most situations wire is the means to conduct electricity from the source to the load. Voltage and current are two important things to address in determining whether or not a wire is appropriate for the task.
The insulation addresses the voltage. Normally hookup wire on battery or lower (50 volts) voltage applications is sufficient - check to be sure if you are connecting to higher voltages.
The wire size addresses the current or amps. Heat and voltage drop are two "current" related issues. If voltage drop is not critical then you can use the smallest wire possible that will not get hot - there are tables that show how much current a wire can carry. As a wire gets smaller the resistance per foot goes up - and the resulting voltage drop along the wire increases. The energy lost to voltage drop is converted to heat but we've already addressed that. If you are concerned about voltage drop then you need to know what current will flow and calculate the voltage drop for the wire size. There are also tables for that.
All of the above presumes that conditions will be "normal" in that the current flow will be what you expect. You need to plan on some not normal things. Current at startup of a motor, for example, is much higher than the run current. If there is a short circuit that results from a component failure or otherwise the current can be much higher. Fuses, fuseable links, circuit breakers, etc were made to address this issue - the let normal currents flow but open the circuit to prevent excess current from flowing.
So, Gene's rule of thumb to not reduce the size is a good guideline unless you 1) know how much current is flowing and 2) are certain that you've addressed the overcurrent problems (short circuits, startup, etc). In your situation you could add a fuse to the points where you connect the smaller wires - the fuse size appropriate for the wire or the weakest link.
Hope this helps.