Setup:  The members of the Zoo Crew have retrieved the Eye of the God artifact and are making their way through a snow-covered mountainous valley.  On the ridge of the valley, multiple minions of the BBEG cast Thundewave on an overhang of snow and ice which breaks and causes an avalanche.  The snow and ice are crashing down becoming larger and larger as it rushes toward the party. 

Question:  In the distance, there is a cave the party can rush to but I want to create a method for how the players can reach the cave while the avalanche is getting closer or overtakes the party.   I need some ideas on how to run such a 'chase' or some type of skill check for the party to outrun the avalanche to reach the cave.  Frankly, any ideas would be appreciated.  

Note 1: I may be taking liberties on how Thunderwave works but I'm playing out the narrative to create the danger of the moment.  

Note 2:  This type of scenario could result in a TPK in which case I have ideas how the party can be 'recovered' by the minions so I'm not worried if it ends in that way.  

Here's what I would do... have everyone make an Athletics Check... sounds like a DC15. Anyone who fails the Athletics Check makes a DC15 Dexterity Saving Throw. The avalanche deals 10d10 bludgeoning damage, and only half-damage on a successful save. Anyone who fails the Saving Throw by 7 or more, or is knocked unconscious by that damage is dragged down by the avalanche and has to deal with whatever the consequences of that are.

Here's what I would do:

Avalanches

The following rules can be used for rockslides as well as avalanches.

An avalanche is a mass of snow and debris falling rapidly down a mountainside. A typical avalanche is 300 feet wide, 150 feet long, and 30 feet thick. Creatures in the path of an avalanche can avoid it or escape it if they’re close to its edge, but outrunning one is almost impossible.

When an avalanche occurs, all nearby creatures must roll initiative. Twice each round, on initiative counts 10 and 0, the avalanche travels 300 feet until it can travel no more. When an avalanche moves, any creature in its space moves along with it and falls prone, and the creature must make a DC 15 Strength saving throw, taking 1d10 bludgeoning damage on a failed save, or half as much damage on a successful one.

When an avalanche stops, the snow settles and buries creatures in it. A creature buried in this way is blinded and restrained and has total cover. The creature gains one level of exhaustion for every 5 minutes it spends buried in the snow. It can try to dig itself free as an action, breaking the surface and ending the blinded and restrained conditions on itself with a successful DC 15 Strength (Athletics) check. A creature that fails this check three times can’t attempt to dig itself out again.

A creature that is not restrained or incapacitated can spend 1 minute freeing another creature buried in the snow. Once free, that creature is no longer blinded or restrained by the avalanche.

To me, simplest would be athletics/acrobatics checks, best of 5 or 7 or whatever and they make it to safety. I’d be willing to give advantage if someone had something relevant, like a ring of free action, or something else that let them ignore difficult terrain. Or if someone makes an argument for a skill or tool proficiency or a class or background feature that would make them somehow better at it. Maybe be let a rogue do it with one fewer success since they have that bonus action dash and so will move faster. Conversely, a halfling or gnome would have to make an extra check since they move slower. I’d go every person for themselves rather than a group check, partly because it seems hard for people running for their lives to stop and help each other, partly because that should let at least one or two of them succeed. I know you’re not worried about tpk, but if only a couple of them make it, there can be some interesting choices for what they do next, and if they can dig out the buried folks before those minions come along.

And while it’s happening, it would be up to you to narrate what a success or failure is like for each character.

You could also incorporate the features of elementals such as the Water Elemental, or treat the avalanche as equivalent to a swarm of elementals of increasing CR as it closes in on the party. Specifically, the Water Elemental has a Whelm ability that can restrain and grapple against a Strength Saving Throw and deals bludgeoning damage.

You could create a [Threat Level] counter, and then have the party make a group check every round to determine how bad the situation gets. If the party has a strong success, then the threat level reduces by 1, to a minimum of 1. If the party fails, the threat level increases by 1. If the part fumbles, then throw an obstacle at them, like a boulder, tree, or an actual elemental.

So, for a possible implementation,

The "Safe Zone" may be 6 rounds away.

Round 1

Phase 1: Party makes an Escape Check. (Adjust Threat Level)
Phase 2: Target [TL] of the party randomly with the Whelm effect.
Phase 3: Party attempts to resolve consequences. (If the party fails to rescue a grappled ally quickly, that player makes the next Escape Check at disadvantage.)

Round 2-5

Repeat Phases 1-3

If the [Threat Level] is greater than the number of players in the party, then trigger a Rescue Mission. Flip a coin for each player to give them a 50/50 chance of being buried. At this point, the snow has settled and the worst has passed, approximately half of the party will have 1~5 minutes to find the other half before they suffocate and die.

With a party of 6, that gives a 1.52% chance of a TPK, and a very high chance of leaving the party beaten, battered, and exhausted.

Thanks, everyone!  These are great suggestions.  I really appreciate you taking the time to work this out.