In this case the emulsification this wasn’t successful due to the mixing of the yogurt with some acid such as lemon. When you add ingredients to any emulsification you must consider how they react or in this case it breaks. Some emulsifiers are more effective than others, and if you use egg yolks they do a particularly good job due to the protein called lecithin.
Emulsifiers are particles that react with both oil/fat and water. Each particle has one hydrophilic (water-friendly) end, and one hydrophobic (oil-friendly) end. The hydrophobic ends attach to the molecules of oil, while their hydrophilic form a water-friendly cocoon around each globule. The hydrophilic ends repel each other, which helps to keep the oil suspended in water.
To create a successful emulsion, you need two things, an emulsifier and force. Force usually in the form blending breaks apart the oil, dispersing it through the surrounding liquid. The emulsifier keeps it from retreating back into itself but test your production unless you can accept surprises.