When in an ongoing combat, I move a friendly unit with a "move to a battlefield" effect to another enemy controlled battlefield. Is the move effect added to the chain immediately? What about attacker and defender designations? Do they get assigned before the current combat ends?

Partly — the move effect is played and put on the chain like any other spell/ability, but nothing gets Attacker/Defender designations at the new battlefield while your current combat is still running.

The spell or ability only proposes the move; the move itself happens when it resolves (normal FEPR order — a Standard Move is banned mid-combat, effects are not). When it resolves, moving into an opponent-controlled battlefield applies Contested there and the next cleanup stages a Showdown/Combat at that battlefield (Rules 323.7–323.8, 452). But a staged combat can't open unless the state is a Neutral Open State, so it just sits staged until the combat you're already in finishes; then the turn player picks which staged combat to begin (Rules 323.12, 463.1).

Designations at the new battlefield are assigned only when that combat opens (Rule 466.2.c) — or, for a unit arriving at a battlefield where combat is already in progress, during the cleanup after the action that brought it there (Rule 466.2.c.3.a). Note the flip side: your unit left the first battlefield, so it loses its designation for the ongoing combat in the next cleanup (Rule 323.2.c).

One caveat if this is a 3+ player game: a battlefield with a combat in progress is an invalid destination for anyone not involved in that combat, so the move instead Recalls (Rule 449.2.a).