Well if you can have it with any, why not with all of them yes
So, 8 (he appeared in that short regarding the creation of the war doctor, so ha, he's modern!), War, 9, 10, 11, 12, and the Caretaker.
The episode concerns the creation (and defeat) of the Valeyard - "an amalgamation of the Doctor's darker sides from between his twelfth and final incarnations."
But ofc the thing is that the "twelfth" Doctor was produced when David Tennant regenerated into... David Tennant. That one.
1 2 3 4 5 6 7 8 WAR 9 10 10 11 12 (order of regenerations to actors)
The same regeneration that incidentally produced two additional Doctors... the one that went off to live with Rose, and the meta-crisis Doctor Donna.
The former is an angsty human (in a show where humans are consistently shown to be war-mongering, ignorant and hateful). And the latter's final memory was begging not to have her mind wiped (essentially destroying her individuality) by the Doctor. Both would have a good grasp on his darker nature.
This episode would show how the two of these are the Valeyard (obviously not revealed at this stage though).
The David-Tennant Valeyard is dying in Rose's universe. Like Doctor Donna, he cannot live properly, and the only thing that was stopping him from exploding originally like Donna was the leftover regeneration energy in his body (he had a much stronger dose than Donna, she began feeling the effects of being a meta crisis nearly straight away).
It's all unfair - how could the Doctor so carelessly create a being, give it the life it always wanted (with Rose), only for it all to be snatched away like this? This must have been foreseen. Tennant-Valeyard falls into despair, and anger.
In the original universe, Donna-Valeyard begins hearing the sound of drums. She's always been hearing them (ever since she was dropped off by the Doctor at the end of her series run). But originally she barely heard the drums at all. Though whenever she noticed alien activity in the news (even if she, being Donna, just brushed it off), the drumming became louder and louder. She has a faint memory of begging a doctor not to kill her, which distresses her.
Tennant-Valeyard comes to the conclusion that the Doctor must fear him - they know each other perfectly, so the Doctor knows the kinds of lengths Tennant-Valeyard would go to in order to survive - he forgets that he came to live in this parallel universe to be with Rose - he surmises this must be a prison, a different reality to be kept away in, where he can't do any harm.
Tennant-Valeyard begins modifying his TARDIS (not included in the final cut, but for anyone who didn't know, when he was dropped off in Bad Wolf Bay, the human-Tennant was given a piece of TARDIS crystal, and told he could grow his own from that) to be able to break back into the original universe, a process that would also destroy his fledgling TARDIS and cause disastrous side effects.
Donna-Valeyard - who is becomingly increasingly distressed by the drumming - now hears the screech and wail of a TARDIS, and begins to change back to the meta-crisis as she remembers what that sound means.
On board current Doctor's TARDIS; it's Capaldi-12. He and <ASSISTANT> have just finished/are starting an adventure, when the TARDIS begins crashing.
12 looks at the monitors - "no, this is impossible, the shields are up!" - what's happening? - an identical TARDIS has entered the universe. It's not only occupying the same time as the 12's TARDIS (as has sometimes happened before when he met previous incarnations) but according to all the equipment it is both the same and a different TARDIS to his own.
The catastrophic side effects of this begin. A blink - and suddenly they're in the previous TARDIS design, with 11 - he looks confused - then another blink, and 12, 11 and ASSISTANT are in 10's TARDIS... then 12, 11, 10 in 9's TARDIS, all in WAR's, all in 8's... each time looking confused and bewildered.
After 8 the blink happens and they are in a dark, twisted TARDIS, its noise screaming and screeching. The Tennant-Valeyard is amongst them for a second, his eyes dark, and then another blink... all but Tennant-Valeyard are in 12's TARDIS. Each of them is quiet, completely freaked out.
The Valeyard-TARDIS (looking dead and battered) materialises in Donna-Valeyard's kitchen. The door opens, revealing darkness and fire from within - DonnaValeyard screams, all her Doctor memories are back, she knows exactly what this is - a force pulls her through the door, it slams shut.
The Valeyard-TARDIS fades away (destroyed)- left standing in the same spot is the Valeyard. An unconscious Donna is by his feet.
The effects of the meta crisis were reversed. Metacrisis-Doctor and Metacrisis-Donna merged back and separated again - all the "human" came back together to make Donna, and all the "Timelord" came back together (all the hateful, twisted Timelord) to make the Valeyard (still looks like Tennant at this stage, but with a permanent scowl lol).
So the Valeyard is basically "uninhibited" Doctor; he has ambition and no scruples about killing.
He plans to steal a TARDIS so he can return to Gallifrey, and then rule over it as he knows he should - he's an amazing Time Lord, quite possibly the best, especially since his hands aren't tied by morality. He creates a crude-ish piece of technology to send out a signal.
The Doctors, all still in 12's TARDIS (they can't return to their own timelines because of aforementioned catastrophe), see the signal, which more or less boils down to "Meet me <here>, my old friend." They know the Valeyard as well as they know themselves; they know his ambition. They all agree that this would be a stupid thing to do, and the best thing would be to make up their own plan for containing the threat.
ASSISTANT wonders aloud why the Valeyard would call his past selves "my old friend". They explain that they've met before. ASSISTANT is still unsure; reinforces the point that both parties know the other perfectly. They agree. She points out the Valeyard must have known that they would never agree to meet him on his own terms. So why bother sending out such a message at all? It only weakens his own position. Maybe that message wasn't for them?
The Doctor(s) panic.
"I know exactly what he's planning!" says 12.
On earth, the Valeyard is at <location>. A red phone box materialises. "Oh good, you finally made it."
Missy exits her TARDIS, smiling, smug, as usual. Thinks that she's talking to the Doctor.
"I'd travel to the end of the universe to see my best friend," she says in her obsessive/mocking voice. "You know that."
"I know," he says, smiling darkly.
"But," she continues as he stands up, "if this is another one of your plans to imprison me, you can think again."
She uses her phone-device-thing, and reveals a massive network of hostages she controls across the entire earth - and that they will all perish automatically if she is apprehended.
She laughs, thinking she has beaten the Doctor - "But my kind, fragile little Doctor... you make it too easy!"
"Doctor?" the Valeyard says, and there is a bang - Missy's eyes widen, and she peers down to the bullet hole in her chest. "Look closer."
A burst of regeneration energy emanates from Missy, but the Valeyard uses a dark sonic device and siphons it from her - beginning his own regeneration cycle, and interrupting hers. On Missy's device, the lives of the hostages begin blinking out, one by one.
The "true" form of the Valeyard appears - he steps over Missy's body into her TARDIS, and it dematerialises.