What process would explain why the dyes spread through the agar?

Agar -- Agar is polymer and mixture of polysaccharides obtained from the cell wall of red algae .

Source of agar -- Gelidium and Gracilaria are the main commercial genera as source of agar .

Properties - Following properties of agar are seen --

Agar is composed of galactose based polymer of agarose and agaropectin .

Agar has similar properties structural and functional like carrageenans .

Agar is separated from red algae by contact with hot water .

Agar affects Ultra violet radiations .

Application of agar -- Agar is worldwide used, about 90% of total agar as food applications and remain used for the production of capsule for medical purposes .

Due to nature of agar that it solubilize in aqueous buffers above 40 degree centigrade .

It become a gel like at - 38 degree centigrade , which are used for separation of nucleic acids .

Agar gel is most common basic media used in form ( i ) For gel-electrophoresis. ( ii )Gel bead chromatography 

Very importantly agar is used as solid media for growth of microorganisms .

 Due to hydrogel nature of agar ,also improves the mechanical property of the drug delivery system .

Methods by which a medium spread in to another -- Two methods occur by which a medium can be solution , solvent having molecules or ions -

( i ) Diffusion 

( ii ) Passive transport

( i )Diffusion -- In this process a substance move from high concentration to an area of lower concentration .This movement continues till a equilibrium establishes .A concentration gradient which is a form of stored energy in form of potential energy used to equalize concentration ,so this process not involves any energy .

( ii ) Passive transport --Diffusion is considered as passive transport  because no energy required for the process .