The laws of kosher require that in addition to not eating milk and meat together, we wait a specified period of time between eating meat and eating dairy.
After Dairy
After eating dairy and before eating meat, eat something pareve, which does not stick to the palate. Then rinse your mouth, or take a drink, and wash your hands. In addition, many have the custom of waiting a certain period of time — a half-hour or an hour. After eating certain hard cheeses, a six-hour waiting period is required (see Why the extended wait between eating "aged" cheese and meat?).
After Meat
After eating meat foods, we wait six full hours before eating any dairy. The six-hour waiting period is standard for all Jews, except those groups which have halachically established other customs.
If a small piece of meat is discovered between the teeth, remove it and rinse the mouth, but an additional waiting period is not required (even if six hours have elapsed since eating meat). If even the smallest amount of food is chewed or swallowed, the full waiting period becomes necessary.
If food is tasted but immediately eliminated from the mouth before chewing or swallowing, then no waiting period is required. One should rinse the mouth well.
Meat and dairy foods may not be eaten at the same meal, even if they are in separate dishes and even if the waiting time elapses.
Curious to learn the reason behind these waiting periods? See Why Do We Have to Wait Between Eating Meat and Milk?
How do we distinguish what sticks to the palate in order to cleans the mouth after dairy and before meat? Is any pareve food acceptable to do so as long as it’s dislodged from the palate and gone after chewing and swallowing?
And does the time in between start right after finishing eating the dairy or only after cleansing the mouth?
No similar thing is mentioned after meat. Is cleansing the mouth unnecessary so after timing 6 hours we can dive into eating dairy?
The Shulchan Aruch (Yoreh Deah 89:2) writes: "Cleaning is done by chewing bread or any other food, except flour, dates, or vegetables, which adhere to the gums and do not clean well. The mouth is then washed with water or wine. This applies only to the meat of domestic or wild animals; but poultry meat can be eaten after cheese without prior cleaning or washing."
So, any foods that do not stick to the gums are fine.
If you find meat between your teeth, you need to remove it, but in general, you do not need to specifically clean your mouth after six hours have passed. See Shach Yoreh Deah 89:7. You do not need to check for meat between your teeth unless you have an area where meat usually gets stuck. See Kaf Hachaim 89:16.