Styjun

A time travel machine arrives in the late 19th century. A man jumps out and makes the machine disappear into the past so he can't be tracked. He then explains that human descendants with huge brains from millions of years in the future have invaded his time (something like the 25th century) and are transporting the population to the future Earth which is in its last days so the invaders can have a more pristine Earth. The man has stolen a time machine to escape.

I read this in 1987 as part of a book of SF short stories. I do not know the name of the collection, the name of the story, or the author.

Sounds like "Worlds to Barter" by John Wyndham (published under the name John Benyon Harris in the May 1931 issue of Wonder Stories, but Wyndham's first published SF story).

A man arrives in a strange machine, then rigs it up so the machine disappears without him:

The stranger leaned over the instrument board, adjusted several dials, tied a loop in the end of my bit of string and slipped it over a small lever. He took as many steps away as the length of the string permitted and gave a jerky pull. . . .

A change came... there was no machine ; before our startled eyes stood only the stranger, the string dangling from his hand. A sigh of relief broke from his lips as he turned towards us.

He reveals he's a refuge from the 22nd century:

"Please let me go my own way. It is a difficult situation, but I hope I shall convince you. Very few men can have had the chance of convincing their great-great- great grandfathers of anything. I am now an anachronism. You see, I was born in the year A.D. 2108, — or should it be, I shall be born in 2108?— and I am— or will be— a refugee from the twenty-second century.

He explains his sending the machine on was because:

"I daren't keep the thing to examine it. It's even betting that the owners had some way of tracing it and that was not a risk worth taking.

He talks about their first sight of a member of the 5022nd century, who does indeed have large brains:

"In height it must have stood about five feet. The head had twice the volume of ours though the enlargement was mainly frontal. The neck was thickened in order to support the weight until the shoulders barely projected. Puny arms ended in small hands of which no finger carried a nail and none was longer than two inches.

And their goal is indeed to trade places with the people of his time:

"'People of the Twenty-Second Century,' the voice began. 'We of the five thousand and twenty-second century offer peace. We come from a period in the world's history which holds no hope for us. We have conquered time that we may gain the Earth. We offer two kinds of peace, one is elimination, the other, submission to our will.'

"'We are not cruel. We do not wish to kill you, our ancestors. Instead, we give transportation — you will exchange your world for ours. We will carry you across the gulf of half a million years to a world in which you, a short-lived race, will be well suited as will your sons and your sons' sons. For us who count our years by thousands as you count by tens, the end is too near. We have broken through time that we may continue our work. Prepare yourselves and your possessions that you may be ready for the time and places we shall appoint.'