A little more on lip trilling. I look at this skill kind of like double- and triple-tonguing in that you literally have to practice this continually and for a long time to get the speed, accuracy, exactness, etc. And to get the "ka" syllable to sound exactly like the "ta" syllable. There are many ways to practice, the Arban book is a good one. I also practice playing just the "ka" syllable on the tonguing exercises. But I digress, this is about lip trills.
Just to show that the process of learning to lip trill takes an abundance of time, like becoming highly proficient at multiple tonguing.
What I use to get my lip trilling up to speed (very fast), and as a daily warmup, is that I start on an F in the staff and trill that (to a Bb) maybe half a dozen times, then go up chromatically and trill each note all the way up to the F above the staff. As you go up, the trill interval gets smaller, and somewhat easier. Then trill back down chromatically. Try using alternate fingerings as well. It did take me some time to really get these up to speed. Eventually you end up being able to go very fast and quite suitable for use in most musical passages calling for trills.
I also work on lip trills above the F above the staff in the same manner, by starting on the F above the staff, and going up to around high Bb. You usually don't see trills this high, but it doesn't hurt to have this ability.
As for the actual slurring of the lip trill, you can do it as an up down up down up down type of action, but I also found something a long time ago that helped me get faster. That was to practice the trill as if it were a triplet with accents on the 1st note of each triplet. In other words, say on a trill from middle Bb to D, think of Bb-D-Bb / D-Bb-D / etc. Group the notes in three's and lightly accent the first note in each group of three. This allowed me to speed up the trill, and when you can do it fast, it won't sound like triplets.