This could be a slightly garbled recollection of "The Ruum", a short story by Arthur Porges, first published in The Magazine of Fantasy and Science Fiction, October 1953, available at the Internet Archive. You might have read it in one of these compilations. A sequel, "A Specimen for the Queen", appeared in The Magazine of Fantasy and Science Fiction, May 1960, also available at the Internet Archive.
The protagonist of "The Ruum", Jim Irwin, is not a convicted criminal; he was prospecting for uranium in the Canadian Rockies, and his relentless pursuer is a "ruum", a robotic specimen collector inadvertently left behind by exploring space aliens:
The cruiser Ilkor had just gone into her interstellar overdrive beyond the orbit of Pluto when a worried officer reported to the Commander.
"Excellency," he said uneasily, "I regret to inform you tht because of a technician's carelessness, a Type H-9 Ruum has been left behind on the third planet, together with anything it may have collected."
The commander's triangular eyes hooded momentarily, but when he spoke his voice was level.
"How was the ruum set?"
"For a maximum radius of thirty miles, and 160 pounds plus or minus fifteen."
There was silence for several seconds, then the Commander said: "We cannot reverse course now. In a few weeks we'll be returning, and can pick up the ruum then. I do not care to have one of those costly, self-energizing models charged against my ship. You will see," he ordered coldly, "that the individual responsible is severely punished."
But at the end of its run, in the neighborhood of Rigel, the cruiser met a flat, ring-shaped raider; and when the inevitable fire-fight was over, both ships, semi-molten, radioactive, and laden with dead, were starting a billion year orbit around the star.
And on the earth, it was the age of reptiles.
Jim finds the ruum's specimen collection:
It was like some enterprising giant's outdoor butcher shop: a great assortment of animal bodies, neatly lined up in a triple row that extended almost as far as the eye could see. And what animals! To be sure, those nearest him were ordinary deer, bear, cougars, and mountain sheep—one of each, apparently—but down the line were strange, uncouth, half-formed, hairy beasts; and beyond them a nightmare conglomeration of reptiles. One of the latter, at the extreme end of the remarkable display, he recognized at once. There had been a much larger specimen fabricated about an incomplete skeleton, of course, in the museum at home.
No doubt about it—it was a small stegosaur, no bigger than a pony!
Fascinated, Jim walked down the line, glancing back over the immense array. Peering more closely at one scaly, dirty-yellow lizard, he saw an eyelid tremble. Then he realized the truth. The animals were not dead, but paralyzed and miraculously preserved. Perspiration prickled his forehead. How long since stegosaurs had roamed this valley?
Jim meets the ruum:
Jim Irwin had once worked with mercury, and for a second it seemed to him that a half-filled leather sack of the liquid metal had rolled into the clearing. For the quasi-spherical object moved with just such a weighty, fluid motion. But it was not leather; and what appeared at first a disgusting wartiness, turned out on closer scrutity to be more like the functional projections of some outlandish mechanism. Whatever the thing was, he had little time to study it, for after the spheroid had whipped out and retracted a number of metal rods with bulbous, lens-like structures at their tips, it rolled toward him at a speed of about five miles an hour. And from its purposeful advance, the man had no doubt that it meant to add him to the pathetic heap of living-dead specimens.
[. . . .]
It ws easy enough to pull ahead. The ruum seemed incapable of increasing its speed. But Jim had no illusions on that score. The steady five-mile-an-hour pace was something no organism on earth could maintain for more than a few hours. Before long, Jim guessed, the hunted animal had either turned on its implacable pursuer or, in the case of more timid creatures, ran itself to exhaustion in a circle out of sheer panic. Only the winged were safe. But for anything on the ground the result was inevitable: another specimen for the awesome array. And for whom the whole collection? Why? Why?
