The apparatus shown was set up.

Some hours later, the water in the beaker had turned blue, and the liquid in the glass tube had moved upwards.
Which processes caused these changes?

Water turned blue - diffusion ; liquid in glass tube moved upwards - osmosis
. Diffusion is the process by which molecules move from a region of higher concentration to a region of lower concentration. This diffusion continues till every point in the water is at a uniform concentration of the blue ink, hence turning the whole water blue.
The concentration of water in the beaker is higher than the concentration of water in the glass tube. By the definition of osmosis, water always moves from an area of higher water concentration to lower water concentration through a semi-permeable membrane. Also, since the solvent molecules inside the tubing are less, water molecules from outside will rush in, leading to a rise in the water level. Therefore, liquid in glass tube moved upwards by the process of osmosis.