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The Cloak of Beards offers a chance to destroy magic items of enemy characters in base to base contact

Both sides are allowed to move characters within a unit prior to combat

Who moves first?

Can the High Elf move next to a Dwarf character only to have the Dwarf move to another spot in the line?

How do these two rules work together?

You did not specify which edition, but according to the BRB from the 6th edition on P 97, characters move into combat within a unit during their own movement phase, so to answer your question, it depends on who's turn it is. You replace an ordinary trooper as specified in the rules.(I interpret that as NOT being a BSB, Musician, or Champion given the rules on page 95) Once in combat a character can't move.

With all of that I would say that a HE character with the COB, can only move in to use the COB if it is out of combat, and a dwarf character has a ordinary trooper in base contact with him. If he is only touching characters, and/or FC models, then the character with COB has to either wait for an opening on the character or get into combat and declare a challenge.(which he can do that turn as he is now in combat) As far as the Dwarf character, if he comes in contact with a HE character with the COB in close combat, then the dwarf is loosing runic items.

I would have to admit given my experience with WFB, it is a VERY rare occurrence where my dwarf characters were only in base contact with all character type models, so I would say that as long as the HE character is out of combat, he has a good chance of neutralizing a runic item.

This is correct the player whose turn it is gets to rearrange his characters then the opponent can move characters forward who are not in combat.

Once engaged they cannot move. But those characters worried about item destroying spells and abilities can try and move away by declaring a challenge. The trick is either to move a champion into a challenge with dangerous target, or move the vulnerable character away into a different challenge.
Whether this works depends on which player issues the challenge, how many characters each side has and whether there is good reason for the character with the dangerous item/ability to not accept and outsource that job to another character.

Characters in a challenge are part of the larger melee but are considered only in base contact with each other while the challenge lasts.

Thanks for the reply
I should have stated - I play 8th edition.

Does that change anything?

Not really.