How did Dukat beam onto DS9?

Question

DS9 "Tears of the Prophets":

(Dax lights a couple of big candles then turns to the Orb box.)

DAX: I'm not sure if I'm doing this correctly. I don't come here a lot. And to be perfectly honest, I feel more comfortable thinking of you as wormhole # aliens, but Kira believes you're much more than that. Maybe she's right, I don't know. But if you are Prophets, and you're listening, I just want to say

(A wind blows through the shrine and puts out her candles. Dukat beams in.)

DAX: Dukat.

(Dax goes for her phaser but Dukat stretches out his arm and the red Pah Wraith energy lifts her off the floor. Then he lets her drop. She lies very still. Dukat goes through the protective forcefield and takes hold of the Orb box. He opens it and the Pah wraith energy rushes into it. The orb turns from gleaming crystal to dull black. He screams and drops to his knees, no longer possessed. Outside the wormhole opens then collapses in on itself.)

Dukat leaves Cardassia while a huge fleet of ships is heading there to attack it. What ship was he on?

How did Dukat beam onto the space station? Where did he come from?

Answer 1

The script indicates that the next time we see Dukat, he's in the Federation shuttle he stole from Sisko.

It seems likely that that's the reason he was able to approach the station without tripping any warning flags:

81. INT. DOMINION BRIEFING ROOM (OPTICAL): Weyoun and Damar are talking to the image of Dukat, on a monitor. Dukat is on the Federation shuttle he stole in "Waltz."