Connecting old iMac to Mac Studio

if you are planning to use your old iMac as a display for the new Mac Studio, stop. It won’t work. Apple Silicon Macs do not support what was known as Target display Mode.

<< I wonder if Migration Assistant would work with an Ethernet connection. (The instructions only mention WiFi.) >>

yes, it absolutely does! Connecting Ethernet cables through your Router works great. Just connecting a cable sometimes has problems starting up, because the Macs will have self-assigned IP address on those Ethernet ports, and 'Discovering' the other Mac can be more complicated under those circumstances.

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Migration to a new Mac:

Move content to a new Mac - Apple support

Transfer to a new Mac with Migration Assistant - Apple Support

Migration Assistant 'takes over' both computers, and takes a surprisingly long elapsed time. First it may need to compute a Spotlight index of the data. Once data transfer begins, it takes a bit longer than a FULL backup, likely all afternoon to overnight. You may want to set this up late in the day and let it run overnight, and be ready for it not to be done by morning.

"the best way" is to use your Time machine backup from the old Mac as the source for Migration Assistant running on the new Mac. USB-2 is as fast as almost every Rotating Magnetic drive, and will not produce a noticeable slowdown doing this transfer.

The way that usually works (but occasionally take several tries) but will seem really slow is using Wi-Fi through your Router.

If you could use Ethernet through your Router to BOTH Macs, that would be much faster. OR, if your old Mac is running 10.12 Sierra or later it can establish an Ad-hoc private Wi-fi connection to the new Mac when placed near the new Mac and both running Migration Assistant.

If your old Mac has no Thunderbolt-3 ports, Thunderbolt Bridge (a direct connection between the two Macs with a ThunderBolt cable) is off the table.

A USB cord sounds like a great idea, but does not work because USB is a local peripheral interface, and a Network interface is required, unless you can make your old Mac ‘look like a drive' to the new Mac.

Target Disk Mode can allow your old Mac to become a Hard drive, and it can be cable-connected to the new Mac. It requires ThunderBolt cable connection, so for a Thunderbolt-2 old Mac, you would need to obtain a Thunderbolt-3 <-> ThunderBolt-2 adapter (US$50) and a Thunderbolt-2 cable.

Transfer files between two Mac computers using target disk mode - Apple Support

Transfer files between two Mac computers using target disk mode - Apple Support

there are several ways, the fastest of which is to store the files on a external drive and connect that same drive later to your new Mac.

Another way, a bit slower:

if you can get BOTH computers onto the Internet at the same time without re-cabling, you already have a data highway you could use. The connection is through the Switch functions of your Home Router. The data are NOT sent onto the Internet, but are transferred locally from computer to computer through your Router.

the connection is enabled with:

system preferences > Sharing > File sharing

"Sharing" in this context means one-way (no reciprocity) connection for making available for selected other users. To get access, a User will need to provide credentials upon making the connection. These are the credentials that would grant them access to those files if entered locally on the SOURCE computer (i.e., the computer holding the original copies to be "shared".)

Once sharing is anbled, connected Servers should show up in Finder windows on the Network Volume. If they do not, be sure to set IPv6 to "link-local only" to allow MacOS 'Discovery' Protocol to find them.

Yet another way, but WAY more expensive for a 'One-off' transfer:

You can connect the computers with ThunderBolt cables. If the old computer has Thunderbolt-2, you will need the US$50 Apple ThunderBolt-3 <-> Thunderbolt-2 adapter AND a ThunderBolt-2 cable to complete the connection.

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it seems like a USB connection SHOULD work, but Alas, it does not. USB is a peripheral interface, and what is required is a Network-capable interface such as Thunderbolt, using ThunderBolt-Bridge as its Network connection method.

I wonder if Migration Assistant would work with an Ethernet connection. (The instructions only mention WiFi.)

The iMac (27-inch, Late 2013) (iMac14,2) has a 10/100/1000 Ethernet port, while the Studio has a 1/2.5/5/10 Gb one. So hopefully they could auto-negotiate a Gigabit Ethernet connection. Not as fast as a Thunderbolt one – but the cabling might be less expensive, and the connection might be less prone to interference than WiFI.