What terminal command or tool do I use to find and install any needed device drivers for several Linuxs installed on a eSATA external hard drive, when that drive is connected to a totally different computer than the PC used for the installs.?

Usually you don't need to install drivers, except in the case when the kernel version of the installed Linux is to old to recognize newer hardware. In that case I would recommend to upgrade to the newest version of that distribution. Keep in mind that you will have problems if you use proprietary video drivers on such a setup and the video hardware changes.
Which distros are you speaking of?

Okay then and thanks. That external drive has these Linux installed; Fedora 15, PCLinux 2010, Mint 9, Kubuntu 10.04, and Mandriva 2010, and the MBR has the grub 2 from Mint 9. Both the older original computer and the newer PC that I have it connected to now have NVIDIA GeForce video cards installed. The PCLinux and Mandriva did not boot up, but gave a error, ( kernel panic; unable to mount the root file system ). The others seemed to have the needed drivers except for maybe the sound.

You have to figure out if it is failing because it can't find the linux install or because of lack of support. All those should be able to boot to safe mode so I'd think it more likely to be a boot issue.

As long as you dont have closed source drivers installed and have the full set of xorg files its not a problem, even if you swap between nVidia, ATI/AMD or intel video.

BTW, even if you do have nVidia cards installed in both computers doesnt mean that you can install the closed drivers. If one of the systems has a nVidia card that uses different drivers, e.g. one card uses 96.XX drivers (geforce 2 and 4), the other uses 175.XX+ (geforce 6XXX+, 'nvidia-current' in at least some distros) then drivers for one card will not work with the other.

Fedora 15 is end of support-
http://lists.fedoraproject.org/piper...ly/003091.html

Mandriva 2010 is end of support-
http://blog.mandriva.com/en/2011/07/...-from-secteam/

Not a good idea to use distros when they have hit end of support.

There's another potential issue. The video drivers may be properly installed, but for some reason the display resolution is not recognized. This is a not-too-uncommon issue with HTPCs connected through either the VGA or hdmi port of an LCD TV. Many of those TVs don't play nice with EDID, and have "virtual" resolutions which aren't their true native resolution (i.e.: advertised as High-Def or "1080p capable" when their true native resolution is 1366x768). Laptop resolutions can be a problem sometimes, too.

Check if the grub entries are correct for Mandriva and PCLinux. An eSata connection is the same as an internal connection. The cable is different, but connected to the SATA port directly. The root lines in menu.lst are probably incorrect. They may use a /dev/sd# type device name whereas the others use an ID based name.

Another possibility is the devices between the different kernels don't match up with the the bios' or grub's view. You can map devices with grubs HD named devices using the device.map file. Sometimes this is needed to get grub-install working. However as you are using grub (and /boot) from a booting distro, this is probably not the case.

Okay thanks, I think the best idea is to install the newest versions of each of those external drive Linux. The most likely issue would be the SATA AHCI drivers since the older PC only had IDE hard drive, and this newer system does have full AHCI SATA set in the BIOS.