Applying an old EFI update on a new version of OSX I have El Capitan running on a MacMini 2010 that I use as a media player/server. The computer used to be my mother's and I have just noticed that she has never applied any of the firmware updates released by Apple (It is 3 versions behind). 

This has created some performance issues (specially with the newly installed SSD), and I would like to apply the upgrades. But the firmware installer refuses to run on El Capitan, saying it is incompatible with this version of the OS. 

Is there any solution for this problem? Can I, in any way, install the EFI updates without formatting the disk and downgrading the OS? -- Let's say by creating a live OS boot disk, etc...
Simple answer is you can check for the most recent here and it looks like yours would be available here. You can even get this info from the ol' stalwart Mactracker.
Without the exact model identifier I'm just guessing, but assuming you have even the later version from that year (MacMini4,1), the most recent publicly available EFI updates Apple released (at least as of 10.10.4) do not include that hardware. I haven't re-verified it as of either 10.10.5 or ElCapGM, but you can rip apart their publicly posted EFI package and see the javascript-based installer logic for choosing what models the package can patch. I published a post about the topic, and although I haven't gotten confirmation from an authoritative source like Apple themselves, one could assume the most recent-looking model being MM51.88Z.0077.B12.1506081728 means you're out of luck with patches that fix the 'thunderstrike' vulnerabilities.  

And regarding ElCap in particular, since it should have the same hardware support as Mavericks or Mountain Lion, you should be able to confirm it's ready to be upgraded with this command: ioreg -l -p IODeviceTree | grep -E 'firmware-abi.*EFI64'