After a careful analysis of the PHB and DMG, I have found the following bits of information, which I believe can be considered together to arrive at a reasonable conclusion.
PHB 181, "Travel Pace":
While traveling, a group of adventurers can move at a normal, fast, or slow pace[...] A fast pace makes characters less perceptive, while a slow pace makes it possible to[...] search an area more carefully (see the "Activity While Traveling" section later in this chapter for more information).
PHB 182, "Activity While Traveling | Noticing Threats":
Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat[...]
While traveling at a fast pace, characters take a -5 penalty to their passive Wisdom (Perception) scores to notice hidden threats.
Notably, there is no mention of the impact of a slow pace vs. a normal pace here. This may be an omission, or it could be that "search an area more carefully" was simply intended to refer to the Track and Forage activities, rather than to searching more carefully for traps and secrets. But let's read on...
DMG 237, "Using Ability Scores | Ability Checks | Multiple Ability Checks":
In some cases, a character is free to [retry a failed ability check]; the only real cost is the time it takes[...] To speed things up, assume that a character spending ten times the normal amount of time needed to complete a task automatically succeeds at that task.
The surrounding context of this excerpt indicates that it should only be applied when the following conditions are true:
- It makes sense to use the same check to repeat the same actions for subsequent attempts.
- There is no penalty for failing the check (lack of the successful outcome is not a "penalty" in this regard).
This appears to be the 5e equivalent to the 3.5e "Taking 20" rule.
D20SRD, "Using Skills | Taking 20":
When you have plenty of time[...], you are faced with no threats or distractions, and the skill being attempted carries no penalties for failure, you can take 20. In other words, eventually you will get a 20 on 1d20 if you roll enough times. Instead of rolling 1d20 for the skill check, just calculate your result as if you had rolled a 20.
Taking 20 means you are trying until you get it right, and it assumes that you fail many times before succeeding. Taking 20 takes twenty times as long as making a single check would take.
Since taking 20 assumes that the character will fail many times before succeeding, if you did attempt to take 20 on a skill that carries penalties for failure, your character would automatically incur those penalties before he or she could complete the task.
The 5e version is a little more forgiving; it only requires 10x the normal time investment rather than 20x, and grants automatic success rather than "the result as if you had rolled a 20".
If the passive Perception rule of "automatically noticing things where
DC <= 10 + Perception modifier whenever the party is not in a hurry" is insufficient (e.g. if you frequently have traps, hidden monsters, etc. with higher DCs), then the "Multiple Ability Checks" rule seems to be the best fit for what your players are trying to do.
Offer your players the following options:
- When moving at a normal pace (30 feet per round), I'll use your passive Wisdom (Perception) scores to see if you notice any hidden monsters or traps. Monsters that aren't hiding will notice you at the same time that you notice them.
- When moving at a slow pace (20 feet per round), your characters are being stealthy. I'll use your Dexterity (Stealth) checks against the Wisdom (Perception) checks of any monsters that are actively searching for you, and against the passive Wisdom (Perception) scores of any monsters that aren't searching. I'll also use your passive Wisdom (Perception) scores to see if you notice any hidden monsters or traps.
- If you want to actively search for hidden monsters/traps as you travel, you have these choices:
- I secretly roll a Wisdom (Perception) check for you every round. This will slow down gameplay as I make all these rolls - after all, if I only rolled when there was something to find, it would be too obvious a clue. And of course, you run the risk that I roll less than 10, and you miss something that your passive Perception would have allowed you to notice.
- You can move at 1/2 your speed (10 feet per round if also being stealthy, otherwise 15 feet per round) to have me roll twice for each Perception check, or 1/3 your speed to roll 3 times, etc.
- You can move at 1/10 your normal speed (2 feet per round if also being stealthy, otherwise 3 feet per round) to represent that you are being extremely cautious and diligent in your searching. At this pace, I'll grant you an automatic success on your Perception checks. This solves the problem of slowing down gameplay, as I'm no longer making rolls, but since your characters will be moving so slowly, you run the risk of experiencing [hunger/thirst/fatigue] before you get to a suitable location to address it, or that [you are discovered by a wandering patrol/the bad guy completes his plans while you're examining floor tiles/the mayor gets tired of waiting and hires someone else to find his MacGuffin].
- And, of course, when moving at a fast pace (40 feet per round), I'll use your passive Wisdom (Perception) scores, with a -5 penalty, to see if you notice any hidden monsters or traps.
 "The result as if you had rolled a 20" is only relevant when there are negative modifiers, and/or very high DCs. Of course, DM discretion should still be used - if you set a check's DC to 30, you probably had a good reason for doing so, and could certainly be justified in rejecting the possibility of automatic success on such a check.