Yes dirty transmission fluid can cause shifting problems with your vehicle and can also cause your transmission to fail.
For your automatic transmission in your vehicle to work properly the transmission needs good clean transmission fluid and also a good clean transmission filter.
The dirty transmission fluid can cause your transmission filter to clog up as well which can then lead to transmission shifting problems.
If you're noticing that your transmission is taking longer to shift then make sure you have enough transmission fluid in the transmission.
Sometimes it's just a matter of adding some transmission fluid because it's low.
Don't drive with low transmission fluid because it can damage the transmission.
If your transmission fluid is dirty or burnt then it needs changed out.