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As stated in this question, you can't move between attacks when casting spells like Eldritch Blast as per RAW.
My question is, if a GM would allow this to happen (cast a beam to a creature, move, and then cast the second beam at the same creature) invoking RAI or just because he feels like it, would that unbalance the game in any way?
Yes, Depending on your definition of 'balanced'
Using a house rule or breaking from RAW in this instance increases the flexibility of the attack [ for example staggering Eldritch Blast attacks]. This increases how powerful spell attacks of this form are because it allows the caster to do things they normally could not.
Say on a given attack the caster can only see one target and kills them with a fraction of the spells capability - but if they could move during the spell cast they could attack more than one target thus preserving all of the potential damage output of the spell.
hypothetical anecdotes like the one I listed above are the reason why the house rule you're describing 'unbalances' or increases the power of 'multi-attack' spells like Eldritch Blast.
That being said - how much more powerful the spell becomes from this house rule depends highly on the encounters that your PC caster experiences and is extremely difficult to say if the house rule is 'broken' or 'overpowered'. Typically this change in spell mechanics would be advantageous but not game breaking.
A martial combatant with extra attack can fire a bow, then move, then fire the bow again. This doesn't overpower combat. From this point of view, allowing casters to move between beams would not break anything.
But...
Imagine a warlock, Ára, who can see one foe, Tinco, but is in total cover from a second foe, Parma. Ára casts eldritch blast, sending one beam at Tinco and killing them. Ára then moves out of total cover to attack Parma, at which point in time Parma casts counterspell. What happens?
Does Parma's counterspell retroactively cancel Tinco's death? If not, that depowers counterspell. If it does, then the people at the table have more work as they "undo" damage and other effects. Rewinding time is a hassle.
If you don't allow counterspell at all in this situation, deciding that it must be cast when the mage is first casting the spell, then spellcasters become extremely powerful. Imagine Ára casting fireball while behind concealment (safe from counterspelling) then moving into view and releasing the fireball.
Additionally, is Ára considered to be concentrating during the move? If they take damage while moving (from a reaction attack, for example) do they have to make a concentration save? If it fails, does that change Tinco's death?
These questions need to be answered at the table before the situation comes up at the table.
