To run about busily in all directions; to be constantly on the move attending to various matters (yojijukugo).
新聞記者は新しいスクープのために東奔西走している。
The reporter is rushing around everywhere to get a new scoop.
引っ越し前は荷造りや手続きで一週間、東奔西走した。
In the week before the move I was running around taking care of packing and procedures.
NPOのメンバーは被災地支援のために東奔西走して物資を集めた。
The NPO members ran around collecting supplies to support the disaster area.
彼は仕事と家庭のことで毎日東奔西走している。
He is running around every day dealing with work and family matters.