Have another look at this
image. Save it and look at it frame by frame if you can.
As in frames 2 and 3, as soon as the dog's particles have passed through the portal, they have acquired a nonzero speed (given the opening presentation, I'm assuming this is a pretty good clip, enough to launch the dog if that's what the physics calls for). This change in speed does not require a force or energy input because we're effectively just looking at things from a new frame of reference. In fact, it may be more appropriate to say that the dog acquired a nonzero speed relative to the blue portal's reference frame as soon as the orange portal started descending (a change which makes portal physics especially impossible due to the crazy amount of energy created). However, if you want to make the dog stop moving with respect to the blue portal now that it's moving with respect to the blue portal, you need some application of force. One solution would be to glue the dog to the platform it was sitting on, so that the tensile strength of the glue would arrest the dog's motion (let's assume that the dog can survive the forces involved physically intact.) But that's not part of the original question, and another force that could provide the deceleration has not been proposed.
In short: all reference frames are valid. Start at the blue portal's POV and see if your assessment makes sense.
Edit: I prefer the blue portal's reference frame because the object in question is entirely on that side of the portal when the debatable physics takes place. If you want to use the orange portal's reference frame for what happens on the other side of the portal, we need to get into how to evaluate what happens on the other side of an accelerating portal. We can invoke General Relativity for that, but it ends up as a more convoluted version of what I've been saying for the blue reference frame.