In the campaign that I just wrapped up, there was a hallway of endless doors. In a previous campaign it dropped the PCs 900 years into the past, They skipped forward through time, but never made it back to the present. In the past they managed to inspire a chivalrous order of knights, ~500 years too early.
they also go a niche dragon god elevated to a full member of the pantheon.
In the last campaign the hallway was broken, but it still allowed traveling 1-3 years into the future. The biggest change they made was too grab the dead body of a friend from another adventuring group off a battlefield and raise him from the dead in the present. Making small careful changes, they discovered that their actions did not change his memories. He kept to vague warnings and eventually snuck off to wait for a crucial point he knew was coming, he didn't want to change the original events too much, until they reached the crisis point he remembered.
I have been sticking to the theory that time travel creates branches in time stream, so paradox is mostly prevented, they just end up in a new reality. That way they can't "Bill and Ted" their way out of problems.
Eventually they skipped over several months, when an enemy destroyed the door behind them.
Three months after the campaign ended the characters plan to get back together and attack an enemy general, as they know precisely where he will be. On one trip into a future battlefield, they found out who was the most important figure in the war, and asked about the last time there was a confirmed sighting of him. (He liked to sneak around)
I went into both games with the assumption that PCs would alter the world, and being Ok with that. I would do it again (although probably not for a few years so it regains its novelty)
The time branches does not really address the grandfather paradox. If a PC killed his own grandfather he would continue to exist, but the new reality branch would have no memories of him.