Play-by-Posts can be a fantastic medium for roleplaying as you can play with people from around the world at any time during the day or night. You can play in your pyjamas, while slurpily eating spaghetti, or while watching a movie. It also doesn't require much time as most play-by-posts have a daily posting rate. In fact, the fastest play-by-posts I've seen had a posting rate of around 8 - 10 posts a day which, at a paragraph apiece, still isn't a huge whack of time from you.
As a Keeper, it can be tricky as you have to respond to everyone's posts and keep track of movement, positions, clues given, dates, times, and all that Jazz over months of gameplay, but as a player, it's not necessarily that much effort.
The trouble is that even at the fastest rate I mentioned, you're still going to only finish that crime scene at the end of a week. At the more usual rate, it could take a month or two. Therefore, it's doubly important to have identifiable motivations and goals. If players don't know what to do next, they will lose interest and start posting at a slower rate which leads to the scenes taking longer which causes a further loss of interest until it peters out.
So what do you do to prevent this?
Easy. Ensure they always have an immediate goal. They've searched the crime scene but don't know where to go next? Have a neighbour poke their head over the fence causing them to have to either flee or question them. Ensure that the crime scene had obvious enough clues so they know where to go next.
Also, don't just assume that the characters have a goal just because they've got clues. A goal is something they are driven to achieve and randomly dropping by a museum because their is a museum receipt in the woman's handbag just might not be on the cards - or all that interesting to them. So take a look at their posting rate and double check for any "Apologies, real life intrudes" posts on the Out of Character threads (always ensure you have one). You can also often check to see when they last logged in to see if they just haven't been present.
If there's no out-of-game reasons, and especially if they're still posting in other games, feel free to inject a stronger goal into the mix. Don't speed up the posting rate yourself as each person should still have the chance to post, but do add something new to the mix. If it's been a week and yours is the last post, then you can certainly put up another brief post that identifies something new. Perhaps the phone rings in the house or a dog barks out a warning or an NPC points out a coincidence. Hell, their kid sister could call up saying there's someone in the house. Anything that could jump start the game and give the characters a pressing reason to continue.
It takes a long time to achieve anything in a play-by-post so don't waste time letting them scratch their heads. Keep 'em motivated! Keep 'em going!