When attempting to move slowly with a game controller (I’m using an xbox 360 controller), while either running a dedicated server or testing with a client, their speed will never interpolate from a lower value to higher speed. speed seems to be clamped at max run speed.
This behavior is defaulted to stock third person template, whether c++ or blueprints.
If you change Max Acceleration in character blueprint, you can alter how quickly characters speed up and slow down.
(You may have noticed that Mixamo characters take a bit of time to speed up and change direction, but Unreal character from starter content doesn’t. This is main reason. If you give them same Max Acceleration value, they will behave approximately same.)
There are actually lots of attributes that affect character movement. I think they come from UCharacterMovementComponent and so you can see them all here:
Its not acceleration, its is you move controller joystick very little, player should move slowly, this happens on host as expected.
If you try this on client, player doesn’t move slowly, he’s automatically running as if you were pushing joystick all way.
That’s a shame. I haven’t tried network features so I can’t really help you. 
Hopefully you’ll get another answer soon!
Hi ,
Thanks for report! I was able to reproduce this issue in 4.8.1, however it appears to have already been fixed in our main internal build. I will watch to make sure this is consistent as we get closer to next major release (probably 4.9). Thanks again!