With spells like Moonbeam or Flaming Sphere, how should we treat the movement of their areas-of-effect when it comes to passing them behind a thick pillar or around a blind corner?
Should we treat any movement of them as if they are being 're-cast' (and so requiring a clear unobstructed view)? Or could a mage presumably will them somewhere out-of-sight?
It is a general guideline that specific rules override general rules in D&D. The general rule is that spells require line of sight and line of effect. However, both of these spells have specific capabilities that override those general rules.
Flaming sphere:
A 5-foot diameter sphere of fire appears in an unoccupied space of your choice within range.
As a bonus action, you can move the sphere up to 30 feet.
Moonbeam:
A silvery beam of pale light shines down in a 5-foot radius, 40-foot-high cylinder centered on a point within range.
On each of your turns after you cast this spell, you can use an action to move the beam 60 feet in any direction.
Both spells note no line of sight is required.
Both note that you can move their effects.
The spell Arcane Eye wouldn't work as it is described in the spell if this interpretation were not true. Arcane Eye describes passing through one inch holes and out the range of sight (either leading out of line of sight or effect). These spells use the same type of language to describe their capabilities as Arcane Eye.
Arcane Eye
You create an Invisible, magical eye within range that hovers in the air
As an action, you can move the eye up to 30 feet in any direction.
If Arcane Eye were limited to Line of Effect and you had to see the destination, then it could no longer be used to scout areas the moment you couldn't see it.
Overly Technical Note #1: One might try to argue that the phrase "there is no limit how far away from you the eye could move" is what allows the eye to move behind barriers. That argument could be challenged with the scenario of an eye moving along the radius of a circle around the player which keeps the eye neither closer nor further away as it moves. DMs embracing that interpretation would have to reject the ability of the eye moving along that path if it goes behind an obstruction because the eye is no further away from the player.
Overly Technical Note #2: One might try to argue that because the Arcane Eye can "see" the destination that this creates a Line of Effect. This is contradicted by precedent as per answers to this question: a spell's Line of Effect does not require Line of Sight unless the spell says so. A spell does, however, require an unobstructed path to the target to have a Line of Effect. In this episode of the Official D&D Podcast, Jeremy Crawford clarifies these rules (at ~34 minutes) by specifically mentioning that a closed, but transparent, window blocks casting. Thus, though a transparent window may give you Line of Sight through the window - you don't have Line of Effect because you do not have an unobstructed line to the target and the result is that the spell is blocked. Thus a spell like Arcane Eye does not gain Line of Effect because it has Line of Sight: it gains Line of Effect because the spell says it does. If DMs stayed with the general Line of Effect arguments for spells, then Arcane Eye would fail. But DMs generally don't make it fail because "specific overrides general." Arcane Eye gains Line of Effect because the spell says it does using the same type of language as Flaming sphere and Moonbeam.
Crawford clearly notes:
The text of the flaming sphere spell explains how the sphere interacts with barriers, creatures, and pits as it rolls around. As the caster, you don't have to see where you move it –January 25, 2018.
So a bit of a fun fact: Neither of the two spells mentioned actually require line of sight.
Flaming sphere:
A 5-foot diameter sphere of fire appears in an unoccupied space of your choice within range.
As a bonus action, you can move the sphere up to 30 feet.
Moonbeam:
A silvery beam of pale light shines down in a 5-foot radius, 40-foot-high cylinder centered on a point within range.
On each of your turns after you cast this spell, you can use an action to move the beam 60 feet in any direction.
All spells, however, require line of effect unless otherwise noted. These two spells are no different, as they each require you to target an area within range.
A typical spell requires you to pick one or more targets to be affected by the spell's magic. A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area of effect.
To target something, you must have a clear path to it, so it can't be behind total cover. (p. 204, PHB)
Because you are moving spells such as these, you are effectively moving the point of origin for the effect. In order to place the point of origin in a new location, you have to target that location- and in order to target that location, you must have a clear, unobstructed line of effect.
One thing to bear in mind, however:
Once a spell is cast, its effects aren't limited by its range, unless the spell's description says otherwise. (p. 203, PHB)
Just because you can't move the center of the effect around a corner doesn't mean that you can't move the effect so that its area hits an area you can't see. You could, for example, move that moonbeam so that the 5-foot area immediately beyond the corner is covered. Depending on the DM, you could even make an argument that you could move the moonbeam or flaming sphere's point of origin beyond the original range... though this particular DM wouldn't buy it.
According to this unofficial tweet by rules designer Jeremy Crawford from January 2018:
The text of the flaming sphere spell explains how the sphere interacts with barriers, creatures, and pits as it rolls around. As the caster, you don't have to see where you move it –January 25, 2018.
In addition, other spells that require you to target a creature can move beyond your sight. Polymorph targets a creature you can see and as long as you have concentration they stay in the form until the duration expires or it drops to 0 hp or dies. The creature can move wherever it wants. There are plenty of other examples of this as well.
