There is inside fighting as in distance (IN fighting) and inside fighting as in being inside both his arms (which is both). As in ..inside their guard and outside their guard inside their elbows or outside their elbows .That deservesd alot of thought and alot of understanding of your limitations and dangers.
Some styles /fighters talk /think in one or the other.
In regards head movment ,your head follows your knee pattern and your feet.
To just stand wide and move your head isnt the deal your looking for when that is mentioned.
IF that occurs you will be timed and flogged and you'll be caught in a low position comming up or in a side postion comming back across with your head.
Luckily most fighters get excited and throw straight shots at a head thats moving sideways so there is usually an out but its your feet and knees that will get you out of there;unless your in there actually hunting down his openings thats another game.
Thinking in reverse if they are low and moving their head side to side, give yourself the correct distance and follow the path of the head when it moves like that and you cant miss cause it's comming back right through the same arch it first took.

This tells you that to use head movment right on the inside is dangerous stuff cause you can get knocked cold when your head moves into a punch comming the reverse way;
So use it by all means but know 'why your useing it' and where your heading to (saftey first).know whats safe ,know whats not.
Then you dont have to over think in the situation cause you have already done that and you know it when your there. So think about it in depth first and then play around with someone and push and test their natuaral reactions in close.
Moving on the inside in both situations ,To do it for advantage you have to also be moving your feet so you bodily move away from the opponents greatest threat to you (which arm is that, in different situations?) or move at least your knees as well so you go under his shots with your head ,go to the outside of his elbow and you are safe for that one move.
For advandced stuff you can go under and out and pull your other foot around behind you as you go and throw over or under the outside his arm depending on his reaction ;that gets you out of the centre infight to the position of still being at full inside range but you have two hands against his one until he moves his centre line back towards you.
You can get away with that easily if you feel his shot comming across and you use the knees and feet to go out and under the shot.
Remember you have two arms and he only has two arms within this set of rules, so use it.
What im saying here is dont get over excited on the inside and imagine threats that arnt there. When you think about it you can land both your wrists down into a mans elbow points and wait for his reation and then move to his reaction and use the openings;thats game and rare to see considering styles but it works cause two meet two and then its their move to react.so you bought a second.
cause he cant hit you until he moves if he doesnt move from it your free to land those uppercuts or short hooks on the inside of his arms.
Inside fighting isnt really for the begginer straight out,but if you think out all the possibilities and know the dangers of some points and the saftey of others you can then play with their balance off their elbows or in clinches and go around the reaction you know you will get as you move into your saftey area and his danger point is when your feet are backing your move (body)and your punch is coming from the blind spot he created on his initial reaction.

A push one way when in a clinch will usually make the opponent shove against it and you knowing this in depth by praticing will help you to move him the way exact way he is then heading anyway! so its easier to move him around.

You have to know already which way he is going to force against your arm push or your weight lean or shove and you use your feet and move to swing him into the same way he will head anyay.Or you go around his arm when it pushes against your glove and you land straight up or into the opening (you knew was comming) cause you created it.
Its up to you to know where to land bodily too; so your safe and he isnt. Thats a thinking fighter one who has ticked all those boxes in and out of the ring.