Are there any good (bench?) multimeters with an input switch?

[n3mmr]

So one can measure several different things without fiddling with leads and contact jacks.

[beanflying]

How many $ do you want to spend 

There is a lot of options.

HP or Agilent 34970A with 20 channel relay cards and 6.5 digit multimeter plugin fitted. Basically 34401A spec with relays. I have a pair in use for high number of inputs and data logging.

Some of the older and current Keithley, HP and I think Fluke Multimeters also have what they call data cards that work a little similar but generally a lower number of inputs. Hewlett Packard 3457A datasheet here shows an example http://www.testequipmentdepot.com/usedequipment/pdf/3457A.pdf

Standalone switch units like the photo below or something like this Valhalla (used in metrology automation) eBay auction: #292937689715 Bigger and clunky but can be added to most meters.

[tooki]

So one can measure several different things without fiddling with leads and contact jacks.

Can you explain your use case more?

But in general, I guess you didn't look very hard, since most models from the major bench DMM makers (Keysight, Keithley, Tektronix/Fluke) have switchable front/back inputs.

Keithley still makes lots of meters that accept input switch ("scanner") cards or switchboxes to do (industrial/lab/factory) testing at high speed — these can scale up to dozens of inputs.

[n3mmr]

So one can measure several different things without fiddling with leads and contact jacks.

Can you explain your use case more?

But in general, I guess you didn't look very hard, since most models from the major bench DMM makers (Keysight, Keithley, Tektronix/Fluke) have switchable front/back inputs.

Keithley still makes lots of meters that accept input switch ("scanner") cards or switchboxes to do (industrial/lab/factory) testing at high speed — these can scale up to dozens of inputs.

When testing say an amplifier design, one needs to check several voltages in various parts of the circuit, to a rather extreme accuracy and having to move probes around just takes a lot of time, and one good bench multimeter might be ok to get, but six or seven will be way too expensive.
I suppose I could build a switch myself.

[tooki]

Keep an eye out for a used Keithley 2700. I've seen those occasionally go for less than a Keithley 2000 (the multimeter it's based on).

[Kean]

If you're doing the measurements and verification manually, then you probably just want to build a manual switch box.

If you're planning to automate it (or may in future) then look at a Keithley (new or used) with scanner card, or maybe a used Agilent 34970A with multi-channel relay card as suggested by beanflying (and what I have).  The Agilent isn't really intended for front panel input selection, but it can be done.  The 34970A also takes 3 cards so you can do different types of measurements with each.  Not sure how easy it is to select between channels on the Keithley's, but I'd sure like to get one to try - in particular the new DMM6500 with the touch screen 

[oPossum]

HP 3488A equipped with one or more 44470A cards.

[tautech]

SDM3055 and SDM3065X can be optioned with 16ch scanner cards but they have lower max voltage and current ratings than the meters themselves, presumably due to creepage clearances on the plugin card.
Check the specs. 

The OP said for measuring amplifiers, if valve, scanner cards might not be a safe option.

[Kean]

HP 3488A equipped with one or more 44470A cards.

Well the 3488A isn't a DMM, just a switching unit intended for use with a separate DMM.

But that reminds me... if you consider getting a 34970A you need to make sure it is one with the DMM option installed (or more precisely, not one with the option excluding the DMM).
Also there are supposedly fake 34970As on the market, so be careful of where you buy - but that is true of all 2nd hand (and maybe even new) TE these days.

[beanflying]

I would be suspect of a chunk of the 34970A cards from China being 'real' may well be clones given the number available without a matching number of 34970A's. I was lucky I got 4 with my pair when I got them locally.

When testing say an amplifier design, one needs to check several voltages in various parts of the circuit, to a rather extreme accuracy and having to move probes around just takes a lot of time, and one good bench multimeter might be ok to get, but six or seven will be way too expensive.
I suppose I could build a switch myself.

Depending on the level of precision you require and what you are measuring be very careful of what switchgear you use. Reed Relays or low EMF ones then switched may be a better option of conventional switches.

[oPossum]

HP 3488A equipped with one or more 44470A cards.

Well the 3488A isn't a DMM, just a switching unit intended for use with a separate DMM.

It is much less expensive than a DMM + switch combo unit and will work with almost any old DMM.