Situation

The party are at an elemental cult's shrine. The cleric contributed to the desecration of the shrine when she stuck a javelin with continual flame burning into the ground (for RP reasons). That makes her the target.

Proposed method to bury / drown / suffocate this cleric

Earth elemental burrows up out of the ground and grapples cleric. Cleric's Strength is 13, Earth Elemental's is 20. This is a standard PHB grapple attempt, not a special grapple attack in the monster's stat block. Assume that the earth elemental is successful in this attempt.

• Moving a grappled creature
  
  

  
  

  When you move, you can drag or carry the grappled creature with you,
  but your speed is halved, unless the creature is two or more sizes
  smaller than you.

  
  

  
  


• Salient stat block issues: EE Burrow movement speed is 30. EE has two slam attacks (under Multi-attack). It can try to grapple her once.
  
  

  
  

  you can use the Attack action to make a special melee attack, a grapple. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.

  
  

  
  

Two different ways this goes down (yeah, pun intended).

1. Elemental begins 5' under the earth's surface, burrows up (moves 5 or 10 feet), grapples cleric, and then moves 20' remaining (divided by two, so 10 feet of movement remaining) under ground. This does not leave a hole, per earth glide's text: the elemental doesn't disturb the material it moves through (MM, Earth Glide)



2. 
   

   Elemental encounters the party, engages, grabs the cleric with a grapple, and then moves 15' straight down under ground.

   
   
   

   In either case, the cleric suffocates until enough rounds have passed to kill her (18 Constitution, she'll last a little while), or, the party figures out a way to dig her out before she suffocates 5', 10', or 15' under the ground. The cleric has no burrowing speed, nor access to earth glide via items, spells, or features.

   
   

Does this work like I think it does?

Can a creature who can enter a terrain that another creature can't, bring that other creature into that terrain? I know a water elemental can do this, but most PCs easily enter into water. Can a earth elemental do this with earth in a similar fashion?

A note on drowning: the picture I had in my head was of the earth elemental bursting up out of the ground, and then pulling the character under ground. This is similar to a sea monster, or a water weird, reaching up to pull a character into the water to drown them. That is where the idea of "drowning" the PC in the earth like that comes from, although it is strictly more of a suffocating, smothering, burying alive in a literal sense.

The earth elemental cannot take others with it using Earth Glide.

This question is actually specifically answered by Jeremy Crawford in Sage Advice! (page 18)

Can an earth elemental grapple with a creature and then pull it underground and leave it there to die?

No. An earth elemental’s Earth Glide is meant to apply to itself only. The elemental doesn’t take other creatures with it when it moves in this way.

But we can also answer the question with a close reading of the ability text.

Earth Glide. The elemental can burrow through non magical,
unworked earth and stone. While doing so, the elemental
doesn't disturb the material it moves through.

This ability allows the elemental to burrow through the ground, but it does not grant this ability to anyone else (like to an adventurer the elemental grapples). So unless the adventurer also has Earth Glide, they have no ability to move through earth & stone, and would therefore be stopped by the ground as soon as the elemental starts pulling them down.

Additionally, because Earth Glide does not disturb the earth, it does not create a tunnel or any kind of space that the elemental could pull an adventurer into, and then collapse. So ultimately, no, your cleric is not at risk of being suffocated underground.

Earth elementals are siege monsters (2x damage to structures), so it would be plausible for an EE to slowly smash a tunnel and then pull a creature down into it. Creatures standing above it would feel it coming, but the elemental could get a couple thin spots in the ground ready to reach up and break under someone.

After doing this, collapsing the tunnel could achieve a burying effect, but it would be much more limited than Earth Glide in terms of how hard it would be for other PCs to reach them and dig them out.