You Do Need power plug adapter while traveling from Swaziland to Mongolia.

Here is Why You Need An Adapter

The outlet and plug type used in Mongolia are C and E and in Swaziland is M. This is how they look like.

Swaziland

In Swaziland, outlets of type M are in use.

Type M

Mongolia

In Mongolia, outlets of type C and E are in use.

Type C

Type E