Did some more searching on archive.org and found a reference to this board in a book called "Upgrading PCs Made Easy" published in 1991.

Also found a magazine review that makes reference to setting a jumper wire and needing a driver configured via CONFIG.SYS. Unfortunately, cannot find the actual drivers or manual for this thing anywhere. 😒

What speed was the computer before, when the 286 had been in the socket?

Disruptor wrote on 2022-12-26, 18:41:

What speed was the computer before, when the 286 had been in the socket?

12 MHz

I also just tried lowering the system speed to 6 MHZ via the speed selector switch, and SysInfo then reports 6 MHz.

So it is tied to system clock speed. I just wonder if there is a way of enabling some sort of multiplier to boost the processor speed of the 386.

Yes, and according to the manual it runs the ISA bus with 12 MHz (page 9, of https://files.support.epson.com/pdf/e3p___/e3p___pg.pdf )
Third Party Option Boards
If you find that some third party option boards do not function properly in the Equity Ill+ at the higher speed, try setting the CPU speed back to 6 MHz or 8 MHz.
When installing a memory expansion option board, make sure that it is a board capable of supporting the higher bus speed (10 MHz or 12 MHz) of the Equity Ill+. Some boards are rated for no higher than 8 MHz.

This section is also interesting:
Software Problems
When installing a copy- protected software package on the Equity Ill+, set the the CPU speed to 8 MHz. This has been found to have fewer conflicts with copy-protection. After the installation, the CPU can be switched back to the higher speed.
If a software package does not appear to be compatible with the Equity Ill+, try switching the CPU speed to 6 MHz and/or 8 MHz before giving up. Some software packages (e.g., Microsoft Chart 1.01 and Think Tank 1.001) have been found to function only at 6 MHz and 8 MHz

Disruptor wrote on 2022-12-26, 18:50:

Yes, and according to the manual it runs the ISA bus with 12 MHz (page 9, of https://files.support.epson.com/pdf/e3p___/e3p___pg […]
Show full quote

Yes, and according to the manual it runs the ISA bus with 12 MHz (page 9, of https://files.support.epson.com/pdf/e3p___/e3p___pg.pdf )
Third Party Option Boards
If you find that some third party option boards do not function properly in the Equity Ill+ at the higher speed, try setting the CPU speed back to 6 MHz or 8 MHz.
When installing a memory expansion option board, make sure that it is a board capable of supporting the higher bus speed (10 MHz or 12 MHz) of the Equity Ill+. Some boards are rated for no higher than 8 MHz.

This is referring to ISA expansion cards though.

In this case, I'm trying to use a processor upgrade. Thus I'm wondering if I need the original DOS drivers for this particular Cumulus 386SX to potentially enable some sort of multiplier to enable higher clock speed.

Unfortunately, I can't find either original drivers nor the manual anywhere online.

Found another magazine blurb about this upgrade. This one is a bit confusing as it says it replaces 286 processors up to 16 MHz. So that does that mean the processor speed of the upgrade is directly tied to the system clock speed (e.g. no multiplier)?

The 386SX on the board is specified to run with up to 16 MHz.
But your 286 motherboard is not.
It runs with up to 12 MHz, but it is overclocking the ISA bus as I have quoted.

It is fine on a 286/16 that runs the ISA bus with 8 MHz using a 1:2 divider.

And I don't give anything up to the "performance gain of 10 percent to 15 percent". It's just a marketing gag.
But, however, you can run software that uses the 386 protected mode.

Perhaps the included driver is nothing else than an expanded memory manager like EMM386.SYS / EMM386.EXE.

With your 2 MB of XMS you may run Windows 3.0 / 3.1 / 3.11 in enhanced mode: WIN /3
real mode (just in Windows 3.0): WIN /R
standard mode: WIN /S
enhanced mode: WIN /3

Disruptor wrote on 2022-12-27, 00:22:

The 386SX on the board is specified to run with up to 16 MHz.
But your 286 motherboard is not.
It runs with up to 12 MHz, but it is overclocking the ISA bus as I have quoted.

I suppose if this particular 386 upgrade board doesn't allow for a speed multiplier for the CPU, my next step would be overclocking the MB itself with a faster crystal.

I have a spare MB for this particular computer I was thinking of using for overclocking experimentation anyway.

And I don't give anything up to the "performance gain of 10 percent to 15 percent". It's just a marketing gag.

Yeah, the other review I posted suggested this particular upgrade board doesn't really make things any faster. Though I suppose I could benchmark the original CPU and this board and see if there is any appreciable difference.

As your motherboard does not divide FSB clock to ISA bus I absolutely do not recommend you to upgrade the board with a faster crystal.
You may damage components and do experience instability.

Disruptor wrote on 2022-12-27, 00:54:

As your motherboard does not divide FSB clock to ISA bus I absolutely do not recommend you to upgrade the board with a faster crystal.
You may damage components and do experience instability.

Correct, that is a risk I am aware of.

This is why I have a second set of hardware (motherboard and RAM/ROM board) for this particular system.

Change the crystal for
14.31mhz operation (28.62)
15mhz
Then 16.66mhz and see if it works

Hopefully your ram is 80ns or faster

Good Luck