From your comment question, here are some high side and some low side switched circuits:

From your subject question, and as has has already been suggested, at first glance it might seem attractive to use a couple of P-channel MOSFETs, as shown below, to switch between the batteries.
Unfortunately, the MOSFET substrate diodes will create a problem when either battery voltage drops about 0.7 volt lower than the other one.
When that happens, the lower voltage diode will no longer be reverse biased and the higher voltage battery will begin pumping current into the lower voltage one when the higher voltage one is turned on.
In the schematic included below I've shown external diodes, for clarity, and have indicated the path the conventonal current will follow, with R1 representing the 10 ampere load you mentioned.
Note that, in this instance, Q1 is completely turned off and Q2 is in saturation.
I've included the LTspice circuit list so you can play with the circuit if you want to.

The same thing happens if you do low side switching using NMOS.
R2 is the 10 ampere load, and R1 represents control stuff that needs to be ON all the time.

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UPDATE:
While it appears that MOSFETS won't work because of interactions between their substrates which allow reverse currents to flow when the MOSFET is in forward cutoff, it seems like a couple of relays, with thousands of megohms between their contacts when they're open, might.
To that end, here's a circuit which seems to work pretty well in a simulator, with one small caveat: during switchover there's about a 3 microsecond load dropout which doesn't affect the circuit's control circuitry, but only the heavy switched current to the load.
With proper "tuning" that dropout can be brought under control and a seamless switchover achieved, but why should I have all the fun? ;)
Here's the schematic, the plot, and the LTspice circuit list so you can play with/optimize the circuit if you want to.

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64 Left 2 WINDOW 3 24 0 Left 2 SYMATTR InstName D6 SYMATTR Value 1N4148 TEXT -2024 576 Left 2 !.tran 30 startup uic TEXT -2024 600 Left 2 !.inc CD4000.lib TEXT -2024 624 Left 2 !.MODEL SW SW (Ron 0.01 Roff 1g Vt 6V Vh 0) TEXT -1224 680 Left 2 ;SPARE TEXT -1912 416 Left 2 ;24V TEXT -776 416 Left 2 ;12V