Without beating too much around the bush:

I know this sounds kind of alienating or unfitting, but I feel this would add content to the game that wouldn't be as rapidly consumed.

Survive, don't die:

Right now, I think there is a huge emphasis on efficient gear and solo play; even some tips like "don't use this or that weapon because it disrupts your teammates as they all mindlessly bash on mobs, with their non-status-inflicting powerhouses" - Because, why would I want to burn an enemy if I can kill it faster?

Really, it's just about the payout: get some crowns, buy new weapon, set your eye on a new piece of gear, rinse and repeat. Do that maybe with company. (Well, at least what it is happening to me right now.)

What the hell am I babbling about?:

Notice the small, blue hit-counter to the right.

Goals:

This IS NOT meant to make the game flashy and over the top spammy.

This IS NOT meant to be forced into everyone's faces, but rather optional.

This is meant to bring some thought on teamwork - How can WE keep the streak alive?

This is meant to encourage the use of fun but not as efficient gear - How can I build on a specific level's theme and my teammates arsenal?

This is meant to make every room and scenario like a puzzle to solve, moving away form a mostly reward oriented gameplay.

Implementation:

REWARDS! - Yes, I contradict myself, but having a "good rank", or rather "surpassing a certain combo-threshold" would put the icing on the cake in form of fire crystals, rarities, prestige, or whatever you may fancy; but that is not important now, because I said we don't want to be greedy gorgos anymore and rather have some fun, dammit!

Those rewards could be given out per depth, tier or strata, depending if it is accumulative per run. Moving on.

STATUS! - Dance around the burning, shocked, stunned, poisoned -man! The driving force of this suggestion is to give additional effects to status, here are some ideas:

• Fire: Most basic, "keep the streak alive" effect. Adds a hit per fire-tick
• Poison: x2 multiplier on afflicted mobs.
• Shock: First "shock-spasm" adds a hit, second one adds 2, and so on.
• Freeze: Thawing (not breaking) deals ~10 hits.
• Stun: Deals additional hits when killed (~15?).
• Curse: Adds hits when the afflicted mob attacks.
• Sleep: Heals mobs so you can keep on racking up the streak. No additional effect(?).

SPECIALISTS! - Attacking a mob with its weakness adds 1 extra hit on attack; increased damage to its family an adds 1 per level.
If I attack a wolver with a flourish that would be 2 hits per attack; with a Wild Hunting Blade it would 4 hits per hit.

"BRING YOUR PROTO-SWORD!" - Of course that would mean the less damage you deal, the more likely you'll have higher combos; at the cost of being able to survive in higher tiers - it's a trade off. Additionally, as Umbra-Lunatis suggested and adapting it to this suggestion, heavy weapons should do additional hits.

OPTIONAL - I wouldn't want to force anyone to deal with this score-hunt, so there should be an option to disable the counter and the game should only bring it up if you surpassed the threshold for the level/stratus/tier. Everything should work just as it has, if you are not interested.

TECHNICAL - I have no idea how hard that would be to code, but I get it may be rather unrealistic to expect that kind of effort at this point. Anyways, the counter should go up any time a mob takes damage, after checking if it fulfills any requirements to increase the count by more than 1.
The counter should be reset to 0 if there was no damage in x seconds. (Maybe 10?)

Maybe a time limit after which max streaks aren't registered would stop parties from abusing some kind of situation indefinitely, if need should arise.

Expected scenarios:

Now that we established the rules, there are many situations, strategies and things that would come to exist:

FUN!? - The urgency to keep the streak alive would most likely ensue in situations where it is thrilling to dash down the corridors of the clockworks, searching for the next opponent, coordinating the team - solving even them dumbest puzzles may be exciting because time might run out.

SQUADS - Division of roles would be more important, maybe now we'd need someone to scout ahead for tricky situations or opportunities to keep the streak going, while the rest finish off the last group of mobs and catches up to the next. And so on.

DIVERSIFY - Maybe we would actually see someone playing in full chromalisk-gear with an amputator, contributing meaningfully during a royal jelly run.

What if:

• ... there was some leader-board, publicly showing your team's high-score at the gate selection?
• ... there was a pickup that would let you teleport to the party-leader?
• ... there was a combo-booster consumable?