There are USB 3.x male to male cables that should work for you. Here's one example. -> https://www.startech.com/Networking-IO/USB-PS2/usb-file-transfer-cable-windows-mac~USB3LINK
I have not used such a cable myself but there are several such cables on the market and it appears that they all use the same chip inside. It sounds like they can be put in a "serial mode" or an "Ethernet mode". Each mode is much like it sounds, a serial mode will look like having two USB to serial adapters back to back, and Ethernet mode will look like two Ethernet adapters back to back. Cables like this are generally preferable to pairing up serial or Ethernet adapters because this is faster, 5 gigabits instead of 100 megabits per second, and is cheaper, because if someone were to track down a couple gigabit Ethernet adapters the cost would be far higher.
Take great care in selecting cables because there are still many USB 2.0 devices like this which will only get you 480 Mbps. Even worse are a number of cables that do not comply with the USB standard and will SHORT OUT THE POWER PINS!
There are passive, and USB 3.x compliant, crossover cables (compliant because they do not connect the power pins) that could work if someone could find drivers. Cables like this -> https://www.datapro.net/products/usb-3-0-super-speed-a-a-debugging-cable.html
Both laptops do not have to be connected by the same means. A Thunderbolt cable between the Mac Mini and one laptop will create a very fast network, up to 20 Gbps with Thunderbolt 2. Cables like this -> https://eshop.macsales.com/shop/Thunderbolt/OWC/Thunderbolt-Cables
Connecting everything by Ethernet is a "safe" choice as setting up a network this way is quite simple, low cost, well documented, etc. All you need is a couple Thunderbolt to Ethernet adapters and a hub, or three Ethernet adapters (one for each computer, plus using the one Ethernet port already in the Mac Mini) and connect the two laptops to the Mini by Ethernet. Thunderbolt to Ethernet adapters like these should do -> https://eshop.macsales.com/item/Apple/MD463LLAU/
Here's what I'd do in this case, Thunderbolt to Thunderbolt from Mini to one laptop and the USB 3.x transfer cable to the other. The Thunderbolt connection should practically set itself up, it's things like this that the port was made for. The USB transfer cable is not something I've tried before but the people that make them say they work. If not then return it and get a gigabit Ethernet adapter instead.
All non-vintage MacBook Air support target disk mode, including the Apple Silicon based MacBook Air released in 2020.
Many vintage and obsolete Air still support target disk mode as well.
Target Disk Mode is not supported on older Macbook Air that lack a Thunderbolt port. The Mini DisplayPort is purely a video out port, and cannot be used for data transfer, despite the identical connector. Since thunderbolt is far faster than gigabit ethernet, it’s preferred due to speed as mentioned in the comments below.
Any MacBook Air that have Thunderbolt 3 or 2 support target disk mode.
For all devices - regardless of ability to boot to target disk mode, an additional supported option for transferring data between two Macs is using a wired connection and Migration Assistant. Since MacBook Air have no ethernet ports, you will need to purchase the Apple USB to Ethernet Adapter for the older model MacBook Air which does not have a Thunderbolt port, an Apple Thunderbolt to Ethernet Adapter for the newer model which supports Thunderbolt or one of the many third party USB adapters.
If you do not want to purchase the adapters, you could migrate the data off the Air using file sharing or have the Mac opened by a service technician to remove the storage and assist with a converter or another Air to house the drive while it's being accessed.
By deleting all files that are non-essential you can speed up the time of transfer, but the USB to ethernet adapter is 100MB speed and likely the fastest you could transfer files on your Air from 2014 or before. Since 2014, much faster WiFi and Ethernet adapters mean faster transfer speeds than before.
Best Answer
You needs to put one of the computer in Target Disk Mode (Press T on startup) and connect another USB-C equipped Mac with a USB-C cable (or USB-C to USB Adapter/Cable for older computer) Note that the USB-C cable that come with your MacBook Pro 2016 does not work.
There are a more complicated method (and this requires a Thunderbolt 3 Cable (or use a Thunderbolt 3 to Thunderbolt 2 Adapter and a Thunderbolt 2 Cable for older computers) and both computer have to be a Mac that support Thunderbolt. But this does not requires Target Disk Mode. Connect both Macs with the Thunderbolt cable (or with the adapter) and go to System Preferences > Network on both Mac. Go to Thunderbolt bridge (add one if it doesn't exist) and there set configure IPv4 to manually on both. Now set Subnet Mask on both to 255.255.255.0, now give two IPs to both computer (i.e. 192.168.2.1 and 192.168.2.2). Now you can use File Sharing (and other shadings) with the two Macs by Thunderbolt when you use the IP address.