There was no magic transportation, of course. France moved troops by land; British Expeditionary Force landed from sea, and then moved to Belgium during August 1914.
The relevant diplomatic documents are available in so-called "The Belgian Grey Book". Here is Russian edition of 1915. And here are English translations available online.
The events were as follows. Germany made ultimatum (also available in "Grey Book" under no. 20) to Belgium on the pretext of future French invasion:
The German Government cannot but fear that Belgium, in spite of the utmost goodwill, will be unable, without assistance, to repel so considerable a French invasion with sufficient prospect of success to afford an adequate guarantee against danger to Germany.
King Albert asked King George by telegram (no. 25) to make "diplomatic intervention... to safeguard the integrity of Belgium".
United Kingdom issued a diplomatic note (no. 28) stating that:
His Britannic Majesty's Government are prepared to join Russia and France, should Belgium so desire, in tendering at once joint assistance to the Belgian Government with a view to resisting any forcible measures adopted by Germany against Belgium, and also offering a guarantee for the maintenance of the future independence and integrity of Belgium
Finally under no. 40 on 4th August 1914 we have letter by Belgian Minister for Foreign Affairs Davignon to British, French, and Russian Ministers:
The Belgian Government regret to have to announce to your Excellency that this morning the armed forces of Germany entered Belgian territory in violation of treaty engagements.
The Belgian Government are firmly deterrmined to resist by all the means in their power.
Belgium appeals to Great Britain, France, and Russia to co-operate as guaranteeing Powers in the defence of her territory.
There should be concerted and joint action, to oppose the forcible measures taken by Germany against Belgium, and, at the same time, to guarantee the future maintenance of the independence and integrity of Belgium.
Belgium is happy to be able to declare that she will undertake the defence of her fortified places.