If a spellcaster has a touch range spell that can affect multiple targets, such as Heroism cast with a level 2 or greater spell slot, can they move between touching the targets of the spell? Or do the targets have to be close enough to be touched at roughly the same time without movement in between?
No voluntary movement is possible while casting a spell, and this includes moving during casting to satisfy a spell's range component. Range and targets are part of casting a spell, and a spell must have valid targets when it is cast, not after. The definition of touch range means that the target must be touched when the spell is cast (PHB p. 202):
The target of a spell must be within the spell's range. […] Some spells can target only a creature (including you) that you touch.
Exceptions to this rule are possible, but almost non-existent. An exception would involve the spell saying explicitly how the touch effect works after casting. For example, Dispel Evil and Good is a touch range spell that has an initial effect that requires a touch as part of the casting, but later allows its effect to be dispelled by a touch after casting, and it says how this touch works explicitly. Effectively, any touch effect that can work after casting isn't part of the touch-range portion of the spell's casting components.
In general though, spells that have a touch-applied effect that you can “hold” will have a range of Self not Touch, and clearly indicate in the spell description how this self-range spell effect can be used later to affect others via touch, and what kind of action it takes.
Casting heroism specifically
Therefore, to cast heroism on multiple targets, you have to gather your allies together within arm's reach first. This makes multi-target heroism more of a before-conflict preparation device, like a “magical pep talk”.
Yes, in the same way that a single Attack Action that encompasses multiple attacks allows movement in the middle of the action, and thus attacks to be made on as many targets as you have attacks if you have enough movement. The action economy is limited, but flexible within its limits. Movement and action can be co-mingled in general, and I see no RAW that would prevent movement during the casting of a spell.