Yogurt is doing several jobs simultaneously in a meatless gravy, and it is also the ingredient most likely to break the sauce. Both facts come from the same protein chemistry.
It supplies body, acid and fat together
Whisked yogurt thickens a sauce without starch, contributes tartness that balances rich fat, and adds dairy richness that vegetarian dishes otherwise have to build separately.
That combination is difficult to replicate with any single substitute, which is why yogurt appears across so many regional Indian gravies.
It also tenderizes vegetables and legumes slightly through its acidity, which shortens cooking for some ingredients and softens the mouthfeel of others.
Heat makes its proteins clump
Milk proteins in yogurt are held in a loose acid-set structure. Additional heat causes them to contract and squeeze out water, which appears as grainy curds in the sauce.
Vigorous boiling is the fastest route to this, and adding cold yogurt straight into a hot pan is the second fastest.
Once broken, a sauce cannot be fully reassembled, though blending it will disguise the texture at the cost of the dish's character.
The techniques that prevent it
Whisking the yogurt smooth before use removes lumps that would otherwise become the seeds of curdling.
Tempering, meaning adding hot liquid to the yogurt gradually before returning it to the pan, brings its temperature up slowly enough for the proteins to cope.
A spoonful of chickpea flour or cornstarch whisked in stabilizes the mixture, and keeping the pan below a rolling boil after the yogurt goes in does most of the rest.
Which American yogurts work
Plain whole-milk yogurt is the closest to what these recipes assume, and its fat content buffers the proteins against heat.
Greek yogurt is strained and thicker, so it needs thinning with water and it curdles a little more readily because its protein is concentrated.
Low-fat and nonfat versions are the most fragile of all, and flavored or sweetened tubs are unusable here regardless of how neutral they seem.
Souring is a separate decision
Yogurt continues fermenting in the refrigerator, so an older tub is noticeably sharper, and some dishes specifically want that sourness.
Where a recipe calls for sour yogurt and only fresh is available, leaving it covered at room temperature for a short period is the traditional adjustment, within the general food-safety guidance for holding dairy at room temperature.
Where the opposite is needed, whisking in a little cream or milk pulls the sourness back without diluting the body of the sauce very much.