I run a gluten free bakery and yes, over mixing is a concern for many cake batters, cookie doughs, pie crusts, etc.
I find, what makes overmixing an issue is the gums used in the recipes.
This being either xanthan or guar.
An over-mixed cake batter will become very stringy and goopy, and will not pour smoothly when run off of a spoon, for example.
Cookie dough, when over-mixed, will become very tough and elastic-like.
Cookie dough should break apart easily when you separate it, but an overmixed cookie dough will stretch when pulled apart...
Kind of like play dough or fondant.
Just like when baking with wheat flour, GF recipes that use gums (so pretty much all of them) yield a tougher, somewhat unpleasant texture.
cooking.stackexchange.com/users/44722/Matthew
If there's leavening in the cake (baking soda or baking powder) that gets activated once incorporated with the rest of the ingredients, and you substantially overmix, you may lose some of its power as you help the gas escape from your batter.
Unless you're whipping it vigorously, you're not going to be bringing enough air into the batter to make up for it.
This is pretty much like letting the batter sit out on the counter for a while before baking, except faster.
cooking.stackexchange.com/users/1672/Cascabel
It depends on what is in the flour.
Xanthan gum's binding power is certainly dependent on the amount of mixing.
More mixing produces a tougher colloid.
As many gluten-free recipes use xanthan gum as a binder, this will indeed be a problem with overmixing.
If you are using a pre-made gluten free flour, you have to read the ingredients.
If these are pure starches only, you can't overmix.
If they have thickening agents, it is possible that overmixing is bad or not, depends on the thickener used in the specific brand.
cooking.stackexchange.com/users/4638/rumtscho
In terms of 'toughness', yes you can mix it a long time.
However, while you are mixing, you are incorporating air into the mixture, so you must take care to not go crazy.
That being said, the main reason to mix a cake until 'just incorporated' is because of gluten development.
I guess what I am saying is, that within reason, you can mix a gluten free cake for quite a while with no adverse affect.
cooking.stackexchange.com/users/3630/mrwienerdog
In New Zealand, We have 2 main g/f flours to bake with.
Both are a combination of rice flour and tapioca among other ingredients.
using these flours it is easy to over-mix baking recipes using these flours and the baking becomes tough and chewy.
The more tender and delicate
you want your cake, the less you mix it in my case.
As long as I don't use rice flour in my custom blended g/f flour mixes, I can mix the batter "until the cows come home".
cooking.stackexchange.com/users/15408/Julie-Anne Whitson
I use premixed GF flour available from most UK stores.
I start off building a thicker batter to start with by withholding the egg white until the end.
You can thin the mix out to make it easier to work in the earlier stages (I mix by hand).
Once the yolk and dried ingredients are combined sufficiently I then fold the whisked whites until almost fully incorporated.
The air trapped in the white also helps with the rise.
cooking.stackexchange.com/users/61651/GazG
If you're making a regular cake, you add all the first ingredients (eggs, butter, sugar, ...) and mix everything well. When you add the last ingredient, the flour, you mix until it's just incorporated, since over-mixing can make the gluten form and you'll end up with a tough cake.  If you work with gluten-free flour, can you mix as long as you want?
Can you overmix a gluten-free cake?
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<cake><flour><gluten-free><mixing>
cooking.stackexchange.com/questions/23165
cooking.stackexchange.com/users/4580/Mien
While gluten free cakes may be less sensitive to overmixing, gums like Xanthan can be affected by the mixing time or leavening like baking soda or baking powder can be partially activated, decreasing their effect. Overmixing is still a problem if the flour contains xanthan gum or another gum.
Overmixing is still a problem due to the gums, like xanthan gum. There is also leavening in the cake that risks overmixing. Overmixing is not a problem because there is no gluten.