I have 2 separate panels with 100 amp. ratings each - one older one fused, one newer with CB's - each with about 10 circuits. Both have bonded neutrals, using a #6 AWG cable and both have neutral and ground buses bonded together. If the transfer switch wire bundle is run into the older panelboard, can wires be pulled to the newer adjacent box, currently joined by 3/4" conduit, in order to power several circuits in the newer panel when the power goes out? I measured just fractional voltage (e.g., 0.012 volts) between the two panelboard neutral/ground buses with all circuits energized. The transfer switch can carry 10 circuits and 30 amps.
The way your 10 circuit transfer switch works is each switch is a single pole double throw (unless 240 then it is double pole double throw). The “common” switch element “load” leg connects to the generator in 1 position and to the service in the 2nd position. For small units wiring is usually remove wire from breaker connect to transfer switch # 1-10 common element or “load” switch leg. If each switch requires power from the service you run a wire from the breaker to the service side of the switch just connected. Run the Generator neutral / ground to the first panel and the second panel will be ok if fed from the first. If the 2nd panel is a separate feed or service not from the first panel you would be creating a ground loop (not allowed). You would need 1 wire from each breaker to the transfer switch and 1 wire from the “load” or common element/ switch leg of the transfer switch to the wire removed from the breaker. If your transfer switch is Buss fed as many larger ones are there would be 1 main breaker powering the transfer switch. Then powering the 2nd panel if fed from the first L1 & L2 must stay the same in both panels (if you put a breaker from L1 on panel 1 and breaker on L2 in panel 2 with a common buss feeder when going back to the service position it will short the main possibly destroying the 2 switches wiring and tripping the main). If each transfer switch is powered by the branch service breaker this is not a problem.