Natural and logical consequences are meant to be the end of a discipline exchange. In the moment and over time, they are meant to curb behaviors.
It is important to note consequences are very rarely meant to be a starting point in the discipline process. There are so many other better places to start. You might think first of being proactive with positive directions and descriptive praise. You might address behaviors with empathy, positive intent, choices, contribution or challenges prior to using consequences. That said, sometimes consequences are a necessary piece.
Natural Consequences
Natural consequences are what just might happen if the child continues the behavior. This sounds like, “if you don’t finish your homework, you might get a bad grade,” and, “if you don’t wear a coat, you might be cold.” These are things that naturally happen in life and without our intervention. While you can state natural consequences to younger children, these start to make sense and work better to curb behaviors somewhere between three-and-a-half to four years old for many children.
The first part of using natural consequences is to state this to your child. The next is to allow them to make a decision and avoid rescuing them if they continue the behavior. Let’s say you are arguing with your five-year-old about wearing their coat outside, and it is cold. You say, “if you go outside like that, you might be really cold.” Child says, “fine,” and opens the door. If you throw their coat on them the second the cold air hits, you will have this battle again tomorrow. Yes, take the coat with you but let the child feel a bit of the consequence. The natural consequence of feeling cold will help to curb the next debate. I am not saying be stubborn and leave the coat home, take it with you, but let the child feel the cold before giving it to them.
Logical Consequences
Logical consequences can be stated in the negative or the positive. A logical negative consequence is stated if there is bad behavior then there’s a bad related consequence such as, “if you leave the toys all over the floor, we are closing the playroom for the afternoon.” A logical positive consequence is stated if there’s good behavior then there’s good related consequence, such as “if you get the toys cleaned up we can have 5 more minutes to play.”
To be fair, your consequence should match your child’s behavior in time, intensity and content. Matching in time means as immediate as possible. For children three-and-a-half years old and younger, it means immediate. Matching in intensity means the level of consequence matches the level of their behavior (not bigger, you are just being punitive). Matching in content means it is on topic with the behavior. If a child is saying mean things to their sibling, a matched-content consequence would be having to play in separate rooms or finding five nice things to say about their sibling. A non-matched consequence would be taking away a TV time or no dessert. The idea is to keep them thinking on topic.
Examples
Your child grabs a toy from a friend.
Natural: If you grab toys, he might not want to play with you.
Logical Negative: If you grab a toy, you may not have a turn with it.
Logical Positive: If you can give it back nicely, I will be sure you have the next turn.
Your child is fighting getting into the car seat in the morning.
Natural: If this takes too long, we might be late, and you might miss centers.
Logical Negative: If you are out of your seat, we aren’t going (only use this one if not going would be a negative to your child, AND you mean it). Smaller ones would be no music or toys in the car if you usually have them.
Logical Positive: If you get in your seat quickly, you can pick the music.
Got a behavior of concern, and you’d like answers? Post them here.
Parenting by Dr. Rene 
What do you do with a 2 1/2 year old who screams/shrieks, especially in the car on a long trip.
You come more than prepared, bring toys and coloring and snacks. Sit in the back with them, bring books to read, have sing-alongs, have finger-puppet shows. Adults, as you can, take turns driving and entertaining the child. If you do allow screentime, you might save this for the car trip.
As far as consequences go this would be turning off music or putting away toys as they are screaming (logical negative) and allowing toys and music as they can be quiet but that by itself at this little age is not likely to work.