When and how can targets be chosen for concentration spells such as Fly or Invisibility?

Question

Spells like fly or invisibility have a casting time of 1 action and a range of touch. The spells are worded so that you touch a creature and they can either fly or become invisible with limits up to the spells effect limit.

Does this mean as long as the caster is concentrating on the spell they can touch up to X targets to affect them while concentrating?

If so can they in the case of invisibility touch a target - and give it invisibility again - that has had the spell end on them due to them attacking?

If not, with a cast time of 1 action, if 2 targets are 15 feet apart can you move from one to the other to touch both while casting the spell?

Answer 1

These spells don't give any ability to the caster (except potentially flying / invisibility)

The spell has one target, so you can affect only one target. Casting the spell at higher levels grants an extra possible terget per extra spell slot level. So two targets at spell slot level 4, 3 at 5 and so on.

Casting time of one action means that you can cast the spell as an Action, it doesn't give you new options for Actions you can take.

Range of touch means that you need to touch the targets, i.e. they must be within hand's reach. This is while you cast the spell, you cannot move during this Action.

Concentration is needed to maintain the effects of a spell where concentration is mandated. It doesn't say anything about what effect is given. In your cases it is the invisibility or the ability to fly that is maintained.

Answer 2

The spells say that they affect a creature (singular)

Invisibility says (emphasis mine):

A creature you touch becomes invisible until the spell ends.

Fly has a similar wording. It does not say "each creature you touch", or "when you spend your action to touch a creature". The spell affects a single creature. The fact that you maintain concentration on it is one of the components of the duration; the invisibility lasts "until the spell ends" and one thing that will end the spell is you losing concentration.

Fly explicitly tells you that it can affect more than one target when you upcast it (emphases mine):

You touch a willing creature. The target gains a flying speed of 60 feet for the duration...When you cast this spell using a spell slot of 4th level or higher, you can target one additional creature for each slot level above 3rd.

Invisibility has a similar wording. Your concentration on the fly spell applies to all the targets; if you lose concentration, the spell ends on all of them at once. Also, note that your selection of targets applies only "when you cast" this spell, not "for the duration". Thus, when upcast, you must select all of your targets as part of the same Cast a Spell action, on the first turn of the spell. You cannot touch one creature on your first turn and later add another creature to the spell (or as you suggest, the same creature multiple times), even if you have upcast it.

If a spell was meant to affect a succession of targets over multiple turns so long as you maintained concentration, it would be worded like the concentration spell vampiric touch (emphasis mine):

Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.

Can you cast the spell on two targets that are 15 feet apart, using 10 feet of movement to move between them and touching them both? That is actually its own question, and there is not site consensus on the answer - so ask your DM!

Answer 3

In the case of both spells, the effect is that the creature you touch becomes invisible or can fly. The effect is that a creature can now fly or be invisible -- the effect is not flying or invisibleness that can be cast about on as many creatures you can touch at once.