In Shmot 4:20, Rashi explains that the donkey which Moshe took on the way to Egypt was the same donkey which Avraham used for the akeda (Binding of Isaac), and this is the donkey of the Mashiach. What is the message the midrash wants to tell us?
Gur Aryeh seems to have a very nice explanation. The link is a Google book, so you will find it on p. 34.
Avraham, Moshe and Mashiach all had an exalted status. They all transcended holiness approaching a Godly level. All 3 people are loftier than time, space and the universe.
The donkey is the only non-kosher animal connected with a mitzvah, namely that it must be redeemed. The Hebrew word, "Chamor" comes from chomri meaning "material". Kings reign over the material world, symbolized by them riding on an animal. Most kings ride on a horse. But these 3 kings, Avraham, Moshe and Mashiach are spiritual kings and it is more fitting for them to ride on the donkey.
Read the rest of the linked commentary. It is wonderful.
The Lubavitcher Rebbe explains that this was actually an answer to Moshe. See an English Essay of it here. Moshe had two arguments why he shouldn't be the redeemer:
He didn't want to exalt himself above his older brother
He realized he wasn't going to be the final redeemer and therefore thought it was a waste of time for him to take the Jews out of Egypt.
G-d's answer/rebuttal was in the donkey. From the Essay:
To answer these complaints, G‑d had Moses ride the donkey that Abraham had readily saddled to fulfill G‑d's command, indicating to him that he, too, should fulfill G‑d's command without hesitation. The fact that this donkey was also the one that the Mashiach will ride indicated that Moses should regard the redemption from Egypt not as a failed attempt to reach the ultimate redemption but as a necessary phase in it. Furthermore, by emphasizing that the Mashiach will be revealed riding specifically on a lowly donkey, G‑d was telling Moses that his humility (in wishing to defer to Aaron) was in fact his prime qualification for the role of redeemer.
The Essay then goes on to bring the Lubavitcher Rebbe's explanation of the different ways Avraham, Moshe, and Moshiach interacted with physicality (As represented by the Chamor/Chomer connection brought by the Gur Aryeh in @DanF's answer), and how that applies to each of us in our own service of G-d. I tried to summarize this but failed miserably, so read it inside.