Gimli's Route to Rivendell

[Tuor in Gondolin]

Related. What would be the way messagesmight be passed between
Moria and Erebor? By bird or dwarf? Perhaps passing men,
or, given the length of such a journey, dwarves taking messages to the Beornings
and then having them give it to a passing traveler to get to
Dale? And would it be a one- or two-way message sysem?

Or perhaps the best, surest way would be to use rangers
(both rangers and dwarves could be of service to the other
and want to keep on good terms).

[Puddleglum]

Whether there were any specific attempts to see why messages from Moria had stopped we don't know (as has been observed). As far as Golin's embassy, however, consider:

• Dwarves are very proud and don't tend to like to ask anyone else for help (especially Elves - even ones they are friendly with). That Dain was finally driven to seek Elrond's ADVICE (as in, "we dwarves are NOT wise enough to figure this out ourselves") is a big clue as to how IMPORTANT this trip was.
• While Dain didn't know what had happened in Moria, he had to suspect foul play. It's hard to imagine any other likely reason for Balin's new realm to go silent. PLUS, Dain knew that Durin's Bane had been there for hundreds of years (it was actually much longer, but Dain may not have known that) and had been single-handedly responsible for driving out a much stronger Dwarvish people than Balin had.
• "Checking On" Moria would have meant more than just waling into Dimrill Dale and gazing at the gates for a while. It would have meant walking INTO Moria through the Front Gate - where they are most likely to be seen if there ARE any Orcs there. IE, if you want to reconnoiter, it'd better be a reconnoiter IN STRENGTH - and even that is chancy.
• Remember, the Fellowship got in from the back door - where the Orcs were less vigilant. Also, they had a Wizard along who was able to take the Balrog out of the mix. Coming in through the Front Door just could not be done secretly.
• And if you DO try to reconnoiter and Durin's Bane (or Orcs) do get you, then you can kiss your embassy to Elrond Bye-Bye.

And that last point is why I think using Gloin's embassy for a scouting trip to Moria would have been foolish. Send another group, maybe - but if asking Elrond's advice is so important (how often to DWARVES ask for advice) then don't risk the Embassy on such a perilous side-trip.

[Inziladun]

Originally Posted by Tuor in Gondolin

Related. What would be the way messagesmight be passed between
Moria and Erebor? By bird or dwarf? Perhaps passing men,
or, given the length of such a journey, dwarves taking messages to the Beornings
and then having them give it to a passing traveler to get to
Dale? And would it be a one- or two-way message sysem?

I can't see Dwarves passing important messages meant for other Dwarves in the hands of one of an alien race.

The ravens, such as Roäc son of Carc could have been an option. In The Hobbit, Roäc said his "people were few", but the ravens lived for more than 100 years, it was said, and under the protection of the new Kingdom Under the Mountain their numbers might have increased.

Then again, if that was the case, I'd wonder why Balin and Co. wouldn't have had a raven or two with them at Moria. If they had, the fate of the colony should have been brought back to Dáin by them.

Originally Posted by Tuor in Gondolin

Or perhaps the best, surest way would be to use rangers(both rangers and dwarves could be of service to the other
and want to keep on good terms).

I wouldn't think that likely due to the fact the Rangers seem to have mostly confined their activities west of the Mountains. Crossing the Mountains to act as errand boys for Dwarves also seems a bit below their pay grade, and would further have reduced the number of Rangers able to guard Bree and the Westlands.

Maybe after all it was just physically journeying Dwarves passing back and forth, possibly not even going all the way to Erebor or Moria, but meeting at pre-arranged places and times.
When the Moria lot didn't make a rendezvous 2 or 3 times in a row, that could have given Dáin a hint that they'd run into Durin's Bane and were beyond help or hope.