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 the higher speed. The speed seems to be clamped at the max run speed.
This behavior is defaulted to the stock third person template, whether c++ or blueprints.
If you change the Max Acceleration in the character blueprint, you can alter how quickly characters speed up and slow down.
(You may have noticed that the Mixamo characters take a bit of time to speed up and change direction, but the Unreal character from the starter content doesn’t. This is the main reason. If you give them the same Max Acceleration value, they will behave approximately the 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 the acceleration, its is you move the controller joystick very little, the player should move slowly, this happens on the host as expected.
If you try this on the client, the player doesn’t move slowly, he’s automatically running as if you were pushing the joystick all the way.
That’s a shame. I haven’t tried the network features so I can’t really help you. 
Hopefully you’ll get another answer soon!
Hi Hitsugen,
Thanks for the 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 the next major release (probably 4.9). Thanks again!