"They are pursuing the criminal."
I like to imagine the difference between -taH and -lI' here in terms of a Klingon cop drama, (or whatever the equivalent may be.)
The two cops are 'pursuing' the criminal, patrolling the city to discover his whereabouts. They don't know how long it will take, or if the criminal is even still there. When they get a radio call asking for their status, the would respond HeSwI' wItlha'taH. Suddenly, they spot the criminal, and give chase! They are about to catch the criminal they've been searching for, so they radio back again HeSwI' wItlha'lI'!
With the caveat that using -taH does not preclude the idea that they are closing in on the criminal. -taH is neutral as to whether you are making progress.
At a very early qep'a', I walked into a room in which a bunch of Klingonists had been playing a music-naming game most of the night. I announced: SubomlI'be'. SubomtaHqu'. We were all new to speaking Klingon in person, so there was a pause while everyone translated in their heads what'd I'd said. Captain Krankor got it first and laughed.