I have changed my DOT3 brake fluid to DOT4 brake fluid. Will it hurt to mix DOT4 fluid with another brand of DOT4 fluid?
tl dr - Mixing Brand X fluid with Brand Y fluid is of no consequence, as long as you are mixing the same grade of fluid (DOT3, 4, 5, or 5.1). In order to be rated at a given grade, it must meet the prerequisites of the grade. This has to do with wet/dry boiling points, not who makes it.
With that said, you can mix DOT3, DOT4, or DOT5.1 together (each of which is glycol ether/borate ether based), but after doing so would need to consider it as the lowest DOTx which is in the mix. Do not try to mix DOT5 in with any of the others, as it is silcone based and there will be adverse effects from it. DOT5 is primarily used by the military, with some application in racing, but is not used very often in consumer vehicles. Silicone based fluid is not compatible with anti-lock braking systems, which is the reason it isn't used very often in consumer vehicles. It is, however, used extensively in extreme-cold environments.
DOTX brake fluid will be the roughtly same for different brands. It will not do any damage to your vehicle when mixing brands. The thing you should keep in mind is that you need to put the same DOTX in your car if you refill. DOT4 will perform a little better than DOT3 as it is more resistant agains higher temperatures. DOT5 will be used in the racing industry and will perform even better at even higher temperatures.
You could compare it to your engine oil. Take for example 10W40 engine oil. These numbers stand for the viscosity of the oil. There are several brands selling these oils and the difference between the brands is that some of them use more or less additives. The more additives there are in your oil, the "cleaner" your engine will be thus they claim. For a more optimal combustion, you need to use more additives. More additives generally cost more. That's also why fuel of brand A is more expensive than brand X.
So overall it doesn't really matter what brands you mix, but it might be better to stick with the same brand. Ironically I've been mixing brands of engine oil over the past three years and never experienced any difference to my car.