The chase takes place on multiple planets and in multiple time-periods. Using a Time-Space Visualizer that they had stolen from the Space Museum, the time-travellers land on a desert planet. Ian and Vicki, exploring, are trapped and hunted by tentacled Mire Beasts. The Doctor and Barbara see on the Visualizer that the Daleks are planning to follow the TARDIS to the Sagarro Desert on the planet Aridus, seize the time-machine and exterminate its crew. They realize that what they have seen is in the near past and the Daleks will be arriving soon.
After a sandstorm, they see the Daleks emerging melodramatically from the sand and begin to dig out the TARDIS, with the help of the Daleks’ native Aridian slaves. Other Aridians find Vicki and Ian, and the time-travellers flee in the TARDIS. The rest of the story is a chase through time and space, with the Daleks beginning fifteen minutes behind them and always closing the gap.
They begin at the the top of the Empire State Building in 1966, then find themselves on board the Marie Celeste in the Atlantic Ocean. Frightened by the arriving Daleks, the crew of the Marie Celeste abandons ship. Then they move to a mysterious old house where both Dracula and Frankenstein’s monster appear. This is really a 1996 future theme park. The robot monsters force the arriving Daleks back to their ship, but Vicki is trapped aboard when they take off in pursuit of the TARDIS. She witnesses the Daleks producing a robot replica of The Doctor (Edmund Warwick). On the jungle planet Mechanus the four travellers are united, but the robot Doctor claims to be the real Doctor, only unmasked when he refers to Vicki as Susan, whereupon the real Doctor pretty much beats him to death with his cane. You’d think Daleks could make a better robot.
In the morning, they realize that a vast white metal city stands over the jungle. The native robot Mechanoids capture them and lock them up with Steven Taylor (Peter Purves), a marooned Earth astronaut. The Mechanoids and the Daleks fight over them with disastrous results. They believe Steven was killed in the collapse of the city. The TARDIS crew find the deserted Dalek time-machine and persuade The Doctor to show Ian how to operate it. Ian and Barbara use it to return to Earth in their own time.
The robot Doctor was played by a reasonably lookalike actor and was not CGI, which was only a gleam in the eye of special effects wizards at the time. The Beatles were supposed to appear as themselves in old-age makeup, but Brian Epstein nixed the idea, so they used film from the BBC show Top of the Pops. Ironically, the Beatles’ performances on the music show were erased, leaving their appearance on Doctor Who the only surviving footage.
There was a great deal of humour in the story, mostly poking fun at the Daleks, and there was a broad parody of an American tourist atop the Empire State Building (also played by Peter Purves) which I’m sure was hilarious to the Brits. There was also a good deal of action at a good pace.
William Hartnell was reportedly genuinely hurt by Jacqueline Hill’s and William Russell’s departure, and the Doctor’s sullen irritation was probably not acting. Ian and Barbara had been the moral centre of the series, because The Doctor was inclined to be amoral, if not callous. They were the heroes, again and again, and made The Doctor a little more human. They were also more disciplined actors, however brilliant Hartnell’s characterization at times, because he was always ad-libbing and going off script, forcing them to scramble to get the story back on track.