Were their mounts sent back from the front lines to avoid enemy fire?

Absolutely!

Just like a de-trucking for a motorized unit, ideally a mounted unit would dismount its main rifle units well behind its (possibly still mounted) reconnaissance screen and advance on foot to engage. By 1936 what few Horse units still existed in European armies were well equipped indeed, and even often were partially mechanized and had draught-pullable anti-tank artillery and anti-aircraft artillery. That being said, despite the modicum of tactical mobility afforded them by horses, they were equipped and trained to fight on foot as well as any infantryman.

Horses would be kept in a laager, and at least a man per squad would remain behind to watch the horses. Think of this like leaving a driver behind in a halftrack. Forward-thinking commanders would have the duty of watching the mount be on rotation, allowing every man to potentially avoid the rigors of combat, esp. if they are feeling ill or tired.

From "Axis Cavalry in World War II:"

A mounted unit's firepower was diminished by one fourth in relation to other units of comparable size, since every fourth trooper was detailed to hold his own and three other horses when the unit dismounted to fight.

This is being made in specific reference German cavalry in the period of 1936-1941; and this means that a trupp's mounts would be watched by at least one man. That means, of a squad of 9-11 men, only 7 to 9 of them would march into combat. Naturally expedient measures were taken:

The horses were sometimes tied together in twos facing in opposite directions; this prevented their running away, and removed the need to have a trooper watch over them.

Both excerpts taken from Page 10. My emphasis.

This was not a practice limited to German units, and was practiced by the Polish, Soviet and the American forces during WWII. Naturally, the nature of the German and Soviet armies (whom retained notable levels of horse-mounted units) means that we have more sources on them. However, the US had one official mounted unit see combat in the Pacific theater, it retained much of the interwar Cavalry (Horse) organization and doctrine, including:

The basic cavalry unit was the eight-man rifle squad. Well into the war it retained an organization...two of the riflemen remained in the saddle as horse-holders when the other six dismounted.

(Page 5, World War II US Cavalry Units: Pacific Theater)

We can see that by the end of the 1920s, it was nearly universal across all European-model armies to have their cavalry fight on foot. This necessitated having an organized, drilled manner in deciding when and where to dismount, how to dismount, and more importantly whom to leave behind to make sure you could mounts to return to.

Sources:

World War II US Cavalry Units: Pacific Theater; Rottmann, Gordon L.

Axis Cavalry in World War II; Fowlder, Jeffery T.