There are a few ways of doing this. Two bars of 6/8 plus one of 2/4 (or 4/8, depending on whether the "four-ness" is really evident) would actually be the most common way, or you could do it as a bar of 12/8 followed by one of 2/4. Either way, an explicit "♪=♪" tempo marking over the first change of meter would probably be a good idea.
I would tend to use quavers (eighths) as the base unit as you are courting very "black" notation with semiquavers. I'm not going to say that a semiquaver denominator is incorrect; it's just perhaps a bit harder to read and count. I'm guessing that what you describe sounds like 4 longish beats followed by two less long beats. (I don't mean a slow tempo, but one kind of beat being half again longer than the other.) I suspect the apparent tempo of the piece will depend on how fast the dotted crochets (quarter notes) sound.
If you want to bring it all into one bar, use something like (12 + 4)/8, make sure your beaming does reflect the meter, and possibly insert a Bartók-style dotted bar line between the triplets and the duplets.