automation - Convert a series of PNGs to an animated gif or video file

I have a series of PNG image files that I want to convert into animation.

What is the best way to convert them into a GIF?

Also I have tried various tools to convert them into a video file (like ffmpeg), however they produce a poor quality video. Using the Video Sequence Editor in Blender produces the desired quality however I do not know how to automate this.

(Using ImageMagick's convert takes too long, is there a way to speed it up?)