For a more obscure example, the Japanese anime Tsubasa: Reservoir Chronicle kinda reminds me of early William Hartnell Doctor Who. You've got a small ensemble traveling between dimensions, which leads to a wide variety of different cities, time periods, and worlds where magic does or doesn't exist. They stay in most locations for about 3-5 episodes, get involved in some local dispute (which tends to also tie into the magic feathers that they're on a quest to find, so shades of the Key to Time arc as well), and then move on to the next adventure, and they never have any idea where they're headed to next. It also does a good job of maintaining the sense of danger that early Doctor Who had.
Thanks for all your suggestions, everyone! Legends of Tomorrow is one of those shows I planned to absorb when I got Netflix, only to find that Netflix UK doesn't have it. I will make sure to see it.
Sapphire and Steel is definitely worth a watch, though you'll probably have to get the DVDs, and despite the timewarping format they are always based in the present day (sort of the point, the threat is other times breaking into now). As with the Trek film trope, serials 2, 4 and 6 are better than 1, 3 and 5, in general opinion.
From that point of view, 4 and 6 benefit from being four partners. 2 is eight, but the slow grind of building horror as the situation becomes the normal background arguably makes it more effective overall.
The other thing worth a mention is that the first serial is the only one that mostly predates the decision to move Sapphire and Steel from a 5.10 children's slot to 7pm, so it's much more child friendly and gentle than the rest.
That didn't stop them in the 1960s with their paper thin budgets. The fact most 2005-revival era DW has stayed on Earth only proves the OP's point and with the most unintentional of ironies.
Something that just struck me is that half the time Sapphire and Steel don't really win; they find a temporary solution, or pay off the threat in a way that makes it Somebody Else's Problem.
It's that the crappy quality VHS transfers they did? I rented those from LoveFilm years back. Apparently they're looking at rebooting The Avengers http://www.denofgeek.com/uk/tv/the-avengers/54563/the-avengers-black-and-dekker-to-pen-reboot-pilot
Joanna Lumley famously played the Doctor in the Children in Need Curse of the Fatal Death sketch but I can’t help but wonder what sort of Doctor McCallum would’ve made. This one is being written by Shane Black of Lethal Weapon/Iron Man 3/Last Boy Scout/new Predator movie. I love his stuff but given the very Americana feel to all his previous films, it does seem an odd fit, though apparently he’s said it will be set in England in the 1960s. I’ve a suspicion it will go the way of his Doc Savage project, though.