I do like to lith print, usually using fomatone, and have tried to lith my handmade emulsions, but no luck with those.
I cannot confirm if the foma liquid emulsion would lith or not (fomabrom doesn't).
There are also other commercial liquid emulsions - some from rollei, and Adox' polywarmtone (and maybe some other as well which I don't know about).
I haven't tried it yet, but I expect Adox PWT will lith nicely (as the original Forte PWT seems to have been an absolutely wonderful lith paper, from accounts online and the two 50x60cm sheets I had to play with a few years back).
An issue with lith printing handcoated emulsions might be that they are not as resistant as commercial papers, so the emulsions might not survive lith processing by themselves. My handmade emulsions do survive it, probably due to the amount of hard gelatin I put into them (but I haven't really tested that). Coating method might also matter.
As for general problems with lith printing, you can look at the documents published by Moersch on their site (the developer they sell is called "easylith").