The goal is to have a realistic animation play whatever direction the character is moving, and whatever speed (below some maximum). The character's feet must never slide across the ground.
Ideally, he should be able to sidestep and walk backwards too.

if you move the model, and the animation just manipulates the mesh, then its up to you and your artist to sync the animation speed and the movement rate. IK may help here. or you can just do the math. the animations moves x frames per stride, where a stride is y meters. the character moves at z meters per second. solve for x.

My animator has provided me with different gait anims - walk, jog, run etc. They need to blend into each other as appropriate when the character accelerates.

when the time comes to switch from jogging to run, you tween from the current jog frame to the first run frame. then loop the run animation. this is one form of animation blending. blending from one animation to the next. the other form is individual animations for different parts of the model, IE run combined with swing sword, for example.

What you are looking for is called Motion Extraction, or sometimes also called Root Motion.

Sometimes it is also referred to Variable Motion Extraction, because the motion is not constant, but defined by some node inside the motion itself.

What you basically do is take a given root node, for example the Hip node, and extract the translation and rotation delta from that for the given frame.

Then you filter that transform, to for example only rotate around the up axis, and capture only movement on the ground plane.

Then instead of applying this transformation to the hip node, you apply it to your actor itself.

Your motion is now driving the position and rotation of your character.

When you blend two animations together, for example if you blend between a walk and run motion, you should calculate the delta movement from each motion, and blend these two delta transforms by the same weight as you blend your motions.

This is the basic idea. There are several things you have to take into account, such as looping and scaling of your character.

Your animations will not be in-place animations anymore, but they have to move away from the origin (in case of moving forward or so).

I don't remember where I read this, but in order to have a run animation blend properly with a walk animation, they both need to be proportinally synchronized. This means, if one of the character's foot is touching the floor at 50% time in the running sequence, that same foot also needs to touch the floor at 50% time on the walking sequence.
This is regardless of these sequences' length.

this is what i call the "neutral frame", "matching frame", or "common frame" method. two animations have a keyframe that "matches" - left foot on ground, for example. and that's the point at which you can switch from one to the next cleanly. using this you can avoid tweening. by having short animations with matching frames, such as the first frame, you can simply sequence animations witthout blending. and its actually a bit more realistic. in real life you drop from run to walk as you forward foot hits the ground, which would be the "matching frame", probably at the beginning of both animations. in real life you don't tween from a run stride to a walk stride in mid-air. if you did, you'd probably break you ankle when you landed! <g>.

yet another technique is to use short "transition" animations. IE something that plays just once for a few frames, at the end of a run loop, and before starting a walk loop, for example. or simply tweens from the current frame to the next "common frame", then switches to the new animation.

i personally try to find a "common frame" pose i can use for all animations, then simply use that as my transition point. also, as i recall, my animation engine automatically tweens from current frame to new frame if required when switching ani's.

due to the necessity of modeling player physics realistically (unless its Mario football <g>), you'll probably want to keep the animations separate from, yet driven by, the physics simulation, IE use in-place ani's, and control stuff with code.

To be honest, using the method of the poster above does not really make nice transitions between walk/run/jog. It might work nice for other cases, but not for locomotion.

You do want to blend here and not just switch instantly between the animations, as that suddenly makes the character go a bit faster, looking unnatural.

What you want are real smooth "analogue" transitions, where you can move the analogue stick slowly up, where it goes faster and faster, without any sort of popping and where the speed of the character goes faster and faster.

It looks way nicer to smoothly increase speed from a walk into jog into run into sprint instead of just switching between them at given speed values.

You need to do some full clip or foot plant synchronization there and use the motion extraction I mentioned before, to get the best results.