I don't remember what issues were that I mentioned in my previous post, but I think it is mostly fixed now.

Michal, is there more here? Something with babel?

I've uploaded new configurations for Fontspec. New approach is used, user comamnds are redefined instead of internal Fontspec macros, as was in the previous version. I hope that it works in all real world examples.

It works with Polyglossia, some minor errors are produced with Babel. I will investigate that.

Here is a small example:

-----
\documentclass{article}
\usepackage{fontspec}
\setmainfont{Latin Modern Roman}
\usepackage{polyglossia}
\setmainlanguage{czech}

\newfontfamily\greekfont{Linux Libertine O}[Script=Greek]
\newfontfamily\cyrillicfont{Linux Libertine O}[Script=Cyrillic]
\setotherlanguages{greek,russian}

\begin{document}
Příliš žluťoučký \textit{kůň}

\fontspec{Gentium}

\begin{greek}
Η Πράγα (τσέχικα: Praha), είναι η πρωτεύουσα και μεγαλύτερη πόλη της Τσεχίας.
Χτισμένη στον ποταμό \textbf{Μολδάβα} (Vltava), στην κεντρική Βοημία, έχει 1,2
εκατομμύριο κατοίκους. Αποκαλείται επίσης «η χρυσή πόλη» και «μητέρα των
πόλεων». Από το 1992, το ιστορικό κέντρο της Πράγας ανήκει στον κατάλογο
μνημείων παγκόσμιας κληρονομιάς της UNESCO.
\end{greek}

\begin{russian}
Пра́га (чеш. Praha [ˈpraɦa]) — город и столица Чехии; административный центр
Среднечешского края и двух его районов — Прага-Восток и Прага-Запад. Образует
самостоятельную административную единицу страны.
\end{russian}
\end{document}
-----

Excellent, of course. Minor points:

1) do you really intend "Michal However" in the ChangeLog,
instead of "Hoftich"?

2) I don't like \batchmode behavior, because I've seen it end up asking for input upon error, without any preceding output. If you want to avoid all the insanely verbose output, I'll insert a shell redirection.

3) The recipe I sent you before for copyright stuff isn't right for brand-new files (sorry; thanks for trying), but rather for messing with Eitan-initiated files. For brand-new files, what's needed is:
% newfile.4ht, generated from |jobname.tex
% Copyright 2017 TeX Users Group
|<TeX4ht license text|>

4) I committed the new .4ht files to TL.

committed new latex.4ht to TL. Thanks much.

LaTeX team released new LaTeX kernel, which uses new font encoding, TU, when Unicode engines are used. TU font encoding selects OpenType fonts by default.

I've tested it some time ago and it worked with tex4ht, but there were some additional changes apparently - because tex4ht now totally crashes with both LuaLaTeX and XeLaTeX. Even when you don't use Fontspec package, which was updated as well.

As a workaround, you can use

\usepackage[T1]{fontenc}

in the premble. It will disable the TU font encoding and the document should compile. But without diacritics.

You can use

\usepackage[utf8]{luainputenc}

with LuaTeX to get Latin diacritics.

Fontspec support is broken totally.