We may eat meat shortly after eating dairy. But meals are never served that way. That is, serving dairy hors d'oeuvre followed by a meat meal. Why not? Is there a rabbinical prohibition against doing that? (It's quite possible to be careful about not mixing, especially with disposable plates and silverware.)
Rema OC 494:3:
ונוהגין בכל מקום לאכול מאכלי חלב ביום ראשון של שבועות ונראה לי הטעם שהוא כמו השני תבשילין שלוקחין בליל פסח זכר לפסח וזכר לחגיגה כן אוכלים מאכל חלב ואח"כ מאכל בשר וצריכין להביא עמהם ב' לחם על השלחן שהוא במקום המזבח ויש בזה זכרון לשני הלחם שהיו מקריבין ביום הבכורים:
The custom in all places it to eat dairy foods on the first day of Shavuos. It seems to me that the reason is similar to taking two cooked foods on Pesach as a remembrance of the Karban Pesach and Karban Chagigah. Similarly, we eat dairy foods and afterwards meat foods, and we need to bring with them two loaves of bread on the table, which is in place of the Mizbeiach; there is in this a remembrance of the Two Loaves that were brought on Shavuos.
Clearly there’s no prohibition, even Rabbinically, of having a dairy course followed by a meat course.
Why isn’t it done year-round? I dunno, just because something is permitted doesn’t mean that you have to do it. Maybe people just want to be safe? Maybe it’s just too much of a hassle? Maybe people just don’t care enough about their milchigs to have them first?
While in practice it does not seem that people necessarily refrain from doing such, there would be a source for such a practice from the Zohar in parshas Mishpatim. Here is the Sulam translation of Zohar ח"ב קכה ע"א:
עוד מצאנו שם, שכל מי שאוכל מאכל ההוא של בשר ושל חלב בשעה אחת, או בסעודה אחת, שהם מתחברים יחד. והוא כמו שאוכל בשר בחלב ביחד
We find there, that anyone who eats a food of dairy and of meat in one hour, or in one meal, that they [are considered to] join together, and it is as if he has eaten milk and meat.
It is an explicit Shulchan Aruch (YD 89:2) that one can meat after dairy (with the exception of hard cheeses). The constraints are that one should clean one's hands and mouth (by eating some bread) as well as drink something.
If one ate cheese it is permissible to eat meat immediately afterwards as long as one examines one's hands, so that one should not have any pieces of cheese attached to them, and if it is night and one is therefore unable to examine them thoroughly one must wash them. One must clean one's mouth and rinse it out. One cleans it by chewing bread, thereby thoroughly cleaning his mouth with it, and one can do this with anything that one chooses, except with flour, dates, or vegetables, because they attach to the palate (the area above the esophagus close to the teeth) and do not wipe well. Afterwords one should rinse ones mouth with water or wine.
It is likely that the sum of all these constraints make it impractical or too error-prone to serve meat after dairy - although it is clearly possible - and as @DonielF wrote is often done on Shavuot.