In Eldritch Horror do the small expansions with no board just add on to the first game?
All of the expansions for Eldritch Horror add content to the base game and require the base game to be played regardless of whether or not they include a new board.
Each expansion with the exception of the latest, Masks of Nyarlathotep, contain content that is usable with just the expansion and the base game and their content is independent of the other expansions (though there are some duplicate components; for example focus tokens).
Masks of Nyarlathotep contains Personal Mission cards for every character in every expansion. The Personal Mission cards are only usable if you have the expansion that includes the character of the Personal Mission. There is also a campaign mode introduced where you can use Elder Gods from other expansions. Everything else in Masks of Nyarlathotep does not require the other Eldritch Horror expansions.