Let's look at the text.

In Luke 1.31-33, the angel speaks of a Messiah who is great, "the son of the most high," and who sits on the throne of David, reigning over the house of Jacob forever with a kingdom that never ends.

In Mary's poem of 1.46-56, she doesn't talk much about the typical messianic expectation aside from an agent of radical social change (v. 51-53), justice (52), and helping Israel (54).

Zacharias (1.68-80) mentions redeeming Israel (1.68-69), rescue from enemies (71, 74, which could also be speaking of radical social change). John Yoder, in his book "The Politics of Jesus," says "the expectation here is not cultic or doctrinal; it is not a religious preoccupation, but a social one."

In Lk. 1.77, Zechariah mentions salvation through the forgiveness of sins as his expectation. Then in 79 he uses metaphors of bringing light in darkness, guidance, and peace.

Simeon, in Lk. 2.30-32, mentions salvation and a light for God to reveal Himself (presumably to a people living in darkness)—the "consolation" of Israel (2.25).

Anna (Lk. 2.38) mentions redemption. What is meant by redemption is not clear, whether political or spiritual or both (compare with Isaiah 40.2).

So let's talk about it some more.

What kind of Messiah were Mary, Zachariah, Simeon, and Anna expecting?

These characters from Luke 1-2 seem to view the Messiah as a political figure who will "redeem Jerusalem," save Israel "from our enemies," and fulfill God's promise to Abraham of a great and mighty nation.